I don't really understand what to do with my description and keywords meta tags. Where do they go in my web page, and what should I put in them? Are they necessary? No one sees them, right?

You are correct. They are not necessary, and no one sees them. No one, that is, except search engines. And some search engines very carefully consider the description and keywords when evaluating where to place your page in search engine results. So let's talk about them. First, your metatags go in the 'head' of the page, which means you should put them somewhere between the 

and

 lines. The two tags should look something like this:

Description

Search engines use description meta tags in two ways, too varying degrees:

1. Occasionally search engines will use your description tag in their display of information about the page, when your page comes up in a search result. This is the only time an actual human reader is likely to see your description. Thus, it ought to be something that is human readable and comprehendable, and - if possible - enticing to the reader to click the link. For example, if you go to Google and search on the phrase "secret word" one of the highest ranking web pages is our Secret Word Game. The description under the link on Google's search results is "Secret Word - Guess letters from the alphabet and determine the computer's secret word." Where do you suppose Google got that text? That's right - it's the meta description tag on our game's web page.
2. Search engines look at the description tag and take not of any important words that show up there; they may help the search engine determine what your page is about, and therefore how to rank it for various search queries. Thus, it's important that your description meta tag contain key information related to the page's topic.

Keywords

The name 'keywords' is deceptive, because keywords are not nearly as helpful as key phrases.  Fill your keywords metatag with phrases you think people might use when searching.  having "webpage design" as a keyword is much better than having "webpage" and "design" as two separate keywords.

Be careful not to repeat words or phrases too much, however, as some search engines will penalize you if they find the same word more than three times in your list.  

2016 Update: It appears that search engines are relying on Keywords less and less when it comes to ranking search results. There's a very good reason for this. Unlike the page title, which appears both in the browser title bar, and the search engine links, and unlike the page description, which also appears in search engine results, the keywords don't appear in a human-readable place at all. This means that dishonest web developers felt comfortable using the keywords as a place to try tricky methods to make their web pages rank higher. Since they're easily spammable, search engines are relying on them less and less. I usually tell my clients to put in a few keywords and phrases that pop quickly to mind, but don't spend too much time sweating over it. Craft good content, and let it speak for itself! 

One Last Piece of Advice

Make sure that the key words and prhases you use in your metatags also show up in the text of your page.  No search engine is going to be impressed if your page title, description, and keywords all have the phrase astronaut training program, but that phrase doesn't show up once in your page content.  When it comes down to it, though the metatags are important, what's actually on the page is the most important of all.