Nodev's guide to game engines

Unity (Engine) https://unity3d.com/
- Language: C# or Javascript
- Pricing: Free if you/your company makes less than $100,000 per year, 2 premium subscription tiers available.
- Very popular engine for creating both 2D and 3D games. It's great for 3D but some consider it overkill for 2D games. It has a lot of built-in tools that make development easier, but some of them are so shitty that you'll probably have to write the feature yourself. There is also an asset store where you can pay money for the shit you're too lazy to do.

Unreal Engine 4 (Engine) https://www.unrealengine.com/
- Language: C++ and/or Blueprint
- Pricing: 5% royalty on whatever a game makes over $3000 per quarter
- A popular engine for big and small developers. Mostly used for 3D and is known for top tier 3D visuals while the 2D functionality is incredibly basic, generally considered overkill for a 2D game. Blueprints are visual scripting system so you can see your disaster code as lines and boxes instead of just words and numbers. Still essentially coding so don't think you're getting off easy. They have a marketplace like Unity's asset store.

GameMaker (Engine) https://www.yoyogames.com/gamemaker
- Language: GML (Proprietary language, kinda similar to C)
- Pricing: ($40)
- This engine only has 2D capabilities, but it's very specialized so it doesn't take you long to have some rectangles moving around on your screen. I'm pretty sure most finished AGDG games were made in this engine. Every platform you want to export to costs extra money. Keep your eyes open for sales.

Godot (Engine) https://godotengine.org/
- Language: GDScript (Proprietary, similar to python), optional C# or C++, plus other options.
- Pricing: Free (Open Source)
- Possible to make 2D or 3D games. The most common complaint about this engine is it's lack of documentation. The payoff for putting up with that is full ownership of your game.

Monogame (Framework) http://www.monogame.net/
- Language: C#
- Pricing: Free (Open Source)
- This one is a framework, so it's not for beginners. Most of the boring stuff is taken care of for you, but some things will require a lot of work. Almost entirely code with no visual feedback unless you use a tool like this (https://www.mapeditor.org/). Plenty of documentation available though which helps a lot. The payoff is full ownership of your game.

SDL (Framework) https://www.libsdl.org/
- Language: Written in C but there are bindings for just about any language
- Pricing: Free (Open Source)
- Simple DirectMedia Layer is a cross-platform development library designed to provide low level access to audio, keyboard, mouse, joystick, and graphics hardware via OpenGL and Direct3D. Extremely portable, and a lot of compatibility with Steam for free, for example if a controller works in Steam, it works in SDL. Tons of documentation and permissive zlib license.

Others to check out:
Love2D, Haxeflixel, Construct 2, SFML, LibGDX