Playing Online Texas Hold'em
First, you'll need to know how to play Texas hold'em. The basic idea is that you are trying to make your best five card poker hand using two cards dealt to you plus a five card board of community cards that everyone shares. You can find the poker hand ranking at any internet poker guide.
There is a betting round when you get your cards, then after the three card flop, then after the one card turn and finally after the one card river. The best hand at the end takes the pot.
Once you've signed up with an online poker site and deposited funds (and be sure to use a sign up bonus), you're ready to play. You may wish to start with a free play money game. In these games, you will find that most players are throwing chips into the pot with reckless abandon, since they can always get more from the site.
For this reason, these games are not games of skill, but you can practice getting into the rhythm of the game here. Once you're comfortable with the mechanics, you can move on to freerolls, free to enter tournaments with valuable prizes for a high finish.
Playing Real Money Online Texas Hold'em
You'll want stakes where a loss will be unpleasant but not devastating. You want the money to matter, but not so much that losing will ruin your whole day.
You should probably start at one level below what you think is your optimal level and see how it goes. The more you play the more you learn and the better you can get, so starting small and moving up slowly is a great way to approach online Texas hold'em.
More on Online Texas Hold'em
If you are not winning right away, don't be discouraged. It's very difficult to be a consistent winner at Texas hold'em. Some basic amateur mistakes include playing too many hands, playing too straightforwardly or bluffing too much.