This, this and this gives a brief overview of the below.
These two are related questions.
Here is a supplement:
When to accept, upvote or downvote:
Accept - If the answer solved the problem in question (you can only accept one answer). This also serves to tell everyone else that the problem has been solved so they don't have to read through it to see if one of the answers answers the question.
If you found the answer to your question yourself, you should post it as an answer and accept it.
Upvote - Indicates good quality, usefulness, etc. It can still be the correct answer but not a particularly good answer or solution, hence no upvotes should be given.
Downvote - Indicates an egregiously sloppy, no-effort-expended post, or an answer that is clearly and perhaps dangerously incorrect.
Why answers should be accepted, upvoted or downvoted:
StackOverflow works with reputation. Reputation is a rough measurement of how much the community trusts you; it is earned by convincing your peers that you know what you’re talking about. This is gained by having accepted answer or upvotes and lost by having downvotes. High reputation gives users additional privileges and can be shown off.
While a "thank you" is appreciated, an upvote and/or accept is appreciated more.
Note that if you accept an answer to one of your questions, this also gives you a small amount of reputation.
How to accept an answer:
At the top left of an answer, you will find something resembling the below:
To accept, click on the arrow:
How to upvote:
In the same place as above, click the up arrow: (the same applies to questions)
How to downvote:
In the same place as above, click the down arrow: (the same applies to questions)
A comment to give new users that don't accept any answers:
Remember to [accept and/or upvote](http://meta.stackexchange.com/a/168143/206447) an answer you found helpful.
Remember to [accept and/or upvote](http://meta.stackexchange.com/a/168143/206447) this answer if you found it helpful.