Well done for asking; my answers to you:
"What's a game engine?"
A software engine is a system of functions and configurations used to make use of hardware to provide consistent functionality. In this case a game engine is a system of functions and configurations used to provide consistent fun.
"Can I make my own Game Engine using Dark Basic Pro?"
Yes. DBP is a programming language shipped with development tools that can build a game engine.
All video games have a game engine; and often combine physics and rendering engines and can be derived from a parent engine as a child. This is the case with the Source engine, which is derived from the GoldSrc engine, which in turn is derived from the Quake engine.
"What advantages do I have for writing my own Engine?"
Many, however I will only touch on a few.
* It is a good way to learn about video game infrastructure; and knowledge of such infrastructure helps you learn how to make solid games
* If you manage to create a darn good engine, those you permit to use it for making games can make do so your way, through your terms; giving you some power to influence the industry
* It is a good way to understand how to make use of a game engine; since it tends to be easier to understand your own game engine
* Others can use your game engine to build games or derived engines; which gives you more exposure and potential
* You get to optimize your engine for your target audience and target hardware
* You get to cater the engine to your style of development. Check out this
2D engine by SelfishGames. Check out my
3D engine here. Check out
evolveds lighting engine here. Different engines for different styles and purposes.
Those a just a few benefits; however note that building one is a long term investment. It will take time, but once complete, it should help you produce what you want much more seamlessly than without it, otherwise it is useless.
"Does DBP have it's own engine I can use? If so, what are they?"
I'm not quite with you on this question.
Chris Tate