How stats work in DanMachi

1. Stat Types
There are five normal stats:
Strength
Endurance
Dexterity
Agility
Magic - (Page 39, Volume 1, LN)
You can gain additional stat types through Development Abilities, I'll talk about this further in section 4.

2. Stat Rankings
Each level of each stat (strength, endurance, agility, magic and so on) has letter ranking grades that correspond with a number, one block of stats for each grade which are as follows:
i 0-99
H 100-199
G 200-299
F 300-399
E 400-499
D 500-599
C 600-699
B 700-799
A 800-899
S 900-999 - (Page 39, Volume 1, LN)
SS 1000-??? (1000 to between 1100 and 1300), minimal information available. - (Page 170, Volume 3, LN)
SSS ???-??? (between 1100 and 1300 to unknown), minimal information available.
2.1. How do you gain stats?
[Status] ——it is also a [God’s Grace].
A god used the sacred text, engraved on the person's back, using their blood as a medium. This is a power that only Gods have and can greatly improve the ability of the target. Of course, there are also things like [Experience Points]. Those are gained through experiencing a variety of events so, just like their name implies, they are based on your own experiences in life.
Of course this is not visible, nor could it be created by hands of a Lower World human. If you need me to explain it, it is equivalent of the history of your own experiences. Gods would take the event hidden in the history, let’s say [Defeating monsters], extract it out, and allow it to become food for growth.
The value of completed events in quantity and quality is the [Experience Points].
To Gods, these are visible and can be controlled. Perhaps it is similar to the ancient rituals, that is after defeating enemies, you would be able to get praises and blessings.
Once the sacred text on you back has a sufficient amount of [Experience Points], a Level Up happens and the ability values rise.
This power is basically what the Gods gave to the Lower World humans. - (Page 36, Volume 1, LN)

3. Why are stats important in the long term?
When you meet your condition for level up, all your stats are added to you directly and directly affect how strong a level up you get and are then reset back to i 0. A higher grade ('A' vs 'B' ranking) will get a better benefit when you rank up, even if it's only the difference between 800 and 799 in stats. The Gods of DanMachi refer to this as "Hidden Parameters". - (Page 28, Volume 4, LN)
In theory, this means that somebody who was S 999 all stats at level 1 then leveled up to level 2 i 0, will be just as strong as somebody who was D 500 all at level 1, E 499 all at level 2 and is now level 3 i 0. In practice, the level 3 adventurer would probably still win though because he would have two Development Abilities compared to the level 2 adventurer's one Development Ability as well as more fighting experience. Please refer to section 4 for more information on this.
3.1. Additional information
The "ideal" scenario is to hit 999 or at least 'S' rank for all of your stats and only then level up because a few levels down the line you'll be much more powerful than everyone else who leveled up with middle of the range stats, it's not that convenient though because stats are incredibly hard to raise as you go along unless you have Bell's OP power he got in episode 1, even Aiz had stats a reasonable amount below Bell's at level 5 after two years of constant adventuring in the deep floors before she hit level 6, not to mention that you don't really get to choose when you level up and pushing yourself harder (in order to gain the higher stat rankings) is liable to qualify you for level advancement in the process so it's kind of a catch 22 since you can't really beat up goblins on the first floor for a hundred years to stat up slowly without leveling up as that's even less efficient than just getting a level up while your stats are only in the 500s.
3.2. Special note about ranks above 'S'
Here's the catch though that the girl didn't mention at the end of episode 8; Bell's agility is actually 'SS 1049'. - (Pages 170 & 251, Volume 3, LN)
If you've been paying attention to what I've been saying up until now, there was previously no known SS rank and stats were long assumed to be capped at 999. - (Page 39, Volume 1, LN)
This shows that there are indeed rank tiers above S rank and it's just that everybody in history before was too much of a scrub to hit that tier so nobody knew about it.
What kind of benefits does having an 'SS' rank attribute after level up give you? It's almost completely cut out from episode 9 but Bell's official (unofficial really) nickname is "World’s Fastest Rabbit", - (Pages 6, 73 & 96, Volume 4, LN)
partially because he just obliterated the fastest previous time of 1 year to go from level 1 to level 2 and did it in a month and a half instead, partly because he's now ridiculously fast when running through the streets of Orario to get anywhere.
Wondering about the 'SSS' ranking on the Stat Ranking list? That's for later down the line after volume 5, even I've not read that far yet and only found out about it from spoiler info while double checking my sources on all of this.

4. Development Abilities
I'm going to mostly be copy pasting what volume 4 says here and then comment on it, as it explains it well:
Development Abilities
It is an ability that is discovered, connecting the existing [Basic Abilities].
The time when it is discovered is during [Level Up]. During every Level Up, there is the possibility of adding it onto the [Status]. [Development Ability] had a different special characteristic compared to the Basic Ability, or rather it was the fruition of a specific career ability and it could be strengthened.
Whether or not a Development Ability would be discovered is affected by the [Experience Points] accumulated repeatedly. The abilities that are developed is related to the actions of the person who received the [God’s Grace].
If there are not a great deal of [Experience Points], then even if they [Level Up], a Development Ability would not be discovered. On the contrary, as long as there were [Experience Points] that met the conditions, then it is possible to discover numerous candidates for the ability. Although every [Level Up] you could only acquire one ability, discovering it was unrestricted.
Development Ability will be shown in the [Status] only through [Level Up]. - (Page 20, Volume 4, LN)
4.1 Additional information
An example of a Development Ability is as follows: [Abnormal Resistance] is an ability that defends against the major symptoms of [Poison]. Although it is not flashy, it is viewed as a treasure by Adventurers, who were disturbed by a variety of abnormal status in the Dungeon. Adventurers, who were often subjected to the [Purple Moth] poison powder on the Dungeon Upper Floors would often earn this ability at a relatively early stage. - (Page 21, Volume 4, LN)
Basically, Your experiences through life on your way to your new level from the previous one can qualify you for an ability to pick upon leveling up.
Get poisoned a lot? [Abnormal Resistance]. - (Page 21, Volume 4, LN)
Killed a lot of monsters in a short amount of time? [Hunter]. - (Page 21, Volume 4, LN)
Do a lot of Blacksmithing? [Blacksmith]. - (Page 104, Volume 4, LN)
And so on. As best as I am able to tell, stat types gained from development abilities do have Rankings (i rank onwards) but do not come with an accurate numerical value. I am not 100% sure yet how they rank up, it may be the normal way with values hidden, it may level up a rank in one go if you meet the conditions, it may even automatically just rise one rank each time you level up, at this point I am not certain of the mechanics of Development Ability rank up yet.

5. Sources
Volumes 1, 3 & 4 of the Light Novel, for info on which pages specifically mention the things I said, check the reference note at the end of most paragraphs which is formatted as " - (Page XX, Volume X, LN).
To get the Light Novel yourself to check the sources you can do so via the translator's blog which is found here:
https://sorotrans.wordpress.com/
Even if you don't care about checking the sources, I highly recommend downloading and reading the LN version anyway, as it is highly enjoyable and both the manga and the anime versions miss out on a *LOT* of information and other enjoyable stuff like the side story of how Hestia came to fall in love with Bell before the events of the first episode.