melody

Good melody writing is something that's really hard to put into words - you need to have a grasp of harmony, nonharmonic tones, how to make something sound interesting and go somewhere interesting, use of rhythm to emphasise certain sounds, etc.
It's actually a subject that I think I could have a lot to say on - I've often held conversations with myself where I explain why I do certain things - but I'd have no idea how to structure it, and it'd be near impossible without audio/visual aids.
Harmony and nonharmonic tones are probably the most important aspects. Things like appoggiaturas, suspensions, passing/neighbour/escape tones pretty much form the basis of all melodic writing. I've spent the last two years at uni studying music where they've taken care to explain each one in detail and have us analyze lots of classical (and some contemporary) examples of melodies and chord progressions, so I'm not sure how I might best compress all that into a few paragraphs :P
One other thing that I think is important is 'creativity' - and what I mean here is the desire to take things in unexpected yet somehow logical directions. For example: if you have a scale in your melody which is leading towards the tonic on the first beat of the bar - don't land it on the tonic as the audience predicts! Land it on the note above (use an appogiatura maybe), or change the chord, or land it on the tonic just after the first beat (use a suspension maybe). Do something that's 'unexpected' but yet fits in to the theme of the piece. Raise the tension and immediately release it.

In fact, forget what I said earlier about harmony forming the basis of interesting melodies, I would say that tension and release forms the basis of ALL music (at least, music that is designed to produce certain ranges of emotions, or certain reactions). In dance music, the tension is actually formed by things like dropping out certain instruments, because the audience has a certain expectation of how the piece will go, and by carefully delaying the 'drop the bass' moment longer and longer, and then the snare starts up (you know what I'm talking about) and EVERYONE knows what's coming, but because it keeps getting delayed and delayed (assuming it's not a badly made song where they just get bored and switch off) the tension and their anticipation just keeps rising. It's like a good orgasm - you get close, then you delay it, then you get close again, then you delay it again, and finally when it happens it's so much better because of the anticipation. But the key here is that the audience should have an expectation, and the composer should abuse this expectation as much as possible!

And in club music, the tension is formed by reduction in sound and timbre. I've been really listening to a lot of club music lately and figuring out what I like about it, and one thing I like is when the composer sets up the full groovy beat early into the song, and then plays around with it. Like, they might have a very complicated beat with lots of different instruments, but they all fit together perfectly. Then, the composer takes away one or two of those instruments, and suddenly the remaining ones don't fit together perfectly anymore! It sounds tense, nervous. The listener knows (or hopes) that at some point in the piece, all the instruments will be back in, but they don't know when. The composer continues playing around, teasing the listener with glimpses of the full beat, often using very compressed sounds which offer just a taste of what the original instruments sounded like. Then there's a similar sort of 'drop the bass' moment but not as blatant, where the listener can feel like the song is just about to build up again to its original glory - and then they wait, and then finally the song is full again! The groove makes sense, the instruments are back to their former bright timbre, everything is happy. I love that moment.

The same applies to classical music, but it's almost always melodic and harmonic. Listeners in classical time knew exactly how the various forms of music went. You can see this tension and release in several of the popular forms - in ternary for example (ABA) where you set up the theme, and then you have a section which is completely contrasting, and it builds up tension because the audience knows the theme will return eventually but not when. Sonata form is probably the most useful form to talk about here - when it was first properly developed in the late baroque early classical periods (I think?) it continued to be used in various forms basically to the present day, because it's just that powerful.

Sonata form at its most basic, a simple extension of ternary in fact, consists of statement of theme (or multiple themes), development of themes, and then finally a restatement of the theme in its original key. It's too complicated to go into full detail of how it works (I've unforgiveably oversimplified it here), but the key thing here lies in the restatement of the theme. As sonata form developed, composers started lengthening the development section more and more and playing around with the audience's expectation of this aspect.

Picture (audiate, rather) a grand, triumphant theme. Now have a development which is sad and mysterious. Is the composer about to come back to the theme? No, false alarm, he's still being sad. Wait I think I hear the lead-up to the theme! This is where knowledge of chord progressions comes in handy - there are certain widely used 'cadences' which are accepted throughout basically the entire classical period from the 17th century to the present day. Now, a good theme will solidify the home key in the audience's mind. The development will typically wander around lots of keys, related or unrelated to the home key. Then, to get back into the home key (the restatement of the theme has to be in the home key in sonata form), the composer uses a particular sort of chord progression. Audiences come to recognise these chord progressions. So they hear the first couple chords, and those with a good ear will think 'Aha, he is returning to the home key!' and will build up anticipation but then he doesn't! He takes it somewhere completely unexpected and has another two or three minutes of development. This really builds tension. Finally, when the home key is arrived at, it is all the more glorious because of the release of all this tension.

Now, I've just gone on a huge ramble on something that isn't strictly speaking all that relevant to melodic writing, but all of these aspects can be applied on a much smaller scale. A melody has an expected form; that is, an audience familiar with these sorts of melodies (which modern audiences will be) will expect it to take a certain form, and this is where nonharmonic tones come in handy. Nonharmonic tones feel like they want to resolve. A leading tone (the seventh of the scale) always wants to rise up to the tonic - there's a whole spiel I can go on about semitone movement being really powerful but that's another story. Appoggiaturas want to resolve. Dissonances want to resolve. The power of these lies in how you choose to resolve them (or not resolve them!) Maybe you choose not to resolve a particular appoggiatura - or maybe you change everything else around it instead! Maybe your leading tone doesn't rise up, or maybe it rises up too much! The third of the chord is always really powerful - your melody might look like it's heading towards the third, but then instead you finish on the fourth and resolve it to the third (the basis of the sus4 chord) or maybe you finish on the fourth and then just leave it there! Other techniques like sequences, cycle of 5ths etc all have an expected form that they take and a comfortable place in classical music. And they're used often for a reason - they're very powerful! But maybe every now and then you can take them in completely unexpected ways. (Not too much, or they'll lose their power when you use them normally!)

For a good example of a sequence used for dramatic effect in a piece that I've actually played myself, see this point in the final section of the 'Prelude' from 'Suite Bergamasque' by Debussy: https://youtu.be/g9F4yAww9H0?t=3m24s
Right as it hits 3:37. That's the spot I'm talking about. That sequence had an expected form and an expected path. Audiences of Debussy's day (and even audiences of our day who have been exposed to classical music) would have fully expected the sequence to lead to the top F, on an F major chord (for those without perfect pitch or who don't know the piece, F major was established as the 'home key' early on, inasmuch as Debussy's music has any home key at all). That's the natural resting point. Composers like Bach and Mozart (Beethoven was actually fairly adventurous for his time but even he would probably have done this conventionally) would have ended that sequence on an F. It would have sounded perfectly fine - the point of a sequence is to build tension, and finishing on an F releases that tension. But that wasn't good enough for Debussy! Not only does he rob the audience of that release, he delays it for three whole bars! The F is not reached until later - and even then it's not portrayed as important, with the top A in the last bar on that page being presented as the 'real' end of that sequence. Doesn't that just sound incredibly expressive though?

For an example of building tension in a classical piece, see Beethoven's 'Eroica' symphony, movement 3: https://www.youtube.com/watch?v=u0eWcb1_uGM (first 6 minutes of the video)
This is one of my favourite pieces and I think it's a good demonstration of tension, despite not being an overtly emotional work. It's not in sonata form, it's in scherzo/trio form. The music is incredibly constrained for quite a while - it's always feeling like it wants to break out, and you feel like it could do it at any moment, and there are several false starts, so that when it finally does about a minute in it's such a great release. Not only that though, but there is an extended trio section in the middle (with the horn chords) which serve as contrast but also to make you forget how the first section went! When the first section comes back in at 4:30, you recognise the tune instantly because it was so prominent at the start, but in addition the 'breakout' moment the first time would (hopefully) have been very memorable, in fact the audience would have forgotten everything else except vaguely how the theme went and the fact that there was a breakout moment at some point. So for almost a minute the audience is sitting on the edges of their seats (or at least I was the first time I heard it!) waiting for this breakout moment to come back. Even though it's completely identical to how the section was played to begin with, it sounds like Beethoven is teasing us by 'delaying' the breakout moment more than we feel it ought to be.

Sorry about this incredibly long and rambling post. It's something I do a lot of thinking about, and it's how I approach new genres of music that I don't 'get' - I try and think of it in terms of this framework of tension and release and see if I can understand it that way.