What is research, specifically in Computer Science?

[This answer is colored by the fact that I'm a theoretical computer scientist.]

As an undergraduate, you are (usually) given crisp, well-formulated problems with definite correct answers that you can reasonably expected to discover in a relatively short time period, using only a very narrow range of knowledge and skills. If you get stuck, your instructor can give you useful hints, because they already know how to solve the problem. Moreover, the instructor can probably explain the complete solution to you fairly quickly (in, say, less than a week).

Research involves vague, half-baked, open-ended problems that may resist progress after months or even years of hard work, regardless of how many classes you've taken or books/papers you've read. Nobody else knows the answer to the problem you're working on (or it isn't really research); nobody can even give you a hint how to proceed. Nobody has a clue what other results or techniques will be helpful, or if you're just going to have to make up a completely new technique on the fly. You don't know if the solution you're aiming for is actually useful or interesting, or even if the problem is formulated so that it has a definite answer. Still, you'd better find something interesting, or you'll never graduate.

If that sounds like fun, research might be for you!

At the risk of being trite, research is about creating new knowledge (that's really the underlying condition for PhD acceptance, did you do something new - and worthwhile, and enough of it). However, like JeffE commented, how this is done in CS varies from area to area, quite significantly.

For my area, it's quite like a lot of mathematics research, it all revolves around developing new theorems about properties of problems or computational models &c. (so you think, then think some more, write it down, realise it's wrong, go back to thinking, talk to someone else, think a bit, write something done... until you have something that is "enough")

For other areas, say data security, research might involve developing and implementing new security protocols, or cryptographic algorithms (though in data security, there's also the mathematical theory part, so it's not all the same).

As for why you do it, that's kind of up to you. At one level you're contributing to human knowledge in a way that will hopefully be useful, however the underlying reason a lot of people go into research is really because they love the area they're in, so the "why" is quite personal. You really do have to love it though, as research is usually slow, misunderstood by others and often disheartening - until you get that moment when you have a result (then you go back to the start).

Where the ideas come from is also a tricky thing. If you're thinking far in the future, they come from you, but this isn't as hard as it sounds, once you start to get into a particular discipline, you (should) start asking questions about why things work certain ways, how they can be improved, how they can be combined with other things. Some of these questions will already have answers, others won't, that's where you start the research. In the short term however, you will likely not need to come up with the overall questions and directions for your research, you will be part of your supervisor's research project(s), and they will give you a starting point. Note that they don't necessarily know how to get the answer either, that's why it's research, a lot of excellent students come undone at this point because there's no definite answer, they can't just check things when they're unsure, so you have to be prepared for uncertainty and incomplete knowledge (its your job to complete it!).

Publishing standards you'll also learn from your supervisor, and from reading papers in your area. You'll quickly figure out how much goes into papers of different types.

And most important, how do I come up with an idea to carry research on?

This, for most people, is a learned skill. Increasingly, Phd programs are where this skill is learned. A non exhaustive skill sets include:

• Having the "right" way to approach an issue as a research problem. Learn to identify what issues are not fruitful research problems.
• A set of ethic/behavior codes specific to research.
• Developing a strong level of perseverance (research is grinding)

Why are you doing it?

Because, in an increasing number of fields, these skills are necessary. Phd is a form of apprenticeship approach to the acquisition of these skills.