Your early play put you behind. Your opponent got two corners. ( He didn't defend them very well, but that is beside the point. ) Corners are the most important part of the board for easy life. So early in the game, both players struggle to get them. Your moves 7 and 15 in particular are just gifts to your opponent because they allow him to respond to your move while taking the corner.
Have a look at this:
http://senseis.xmp.net/?CornersThenSidesThenCenterSince so many people play in the corners early in the game, certain patterns of corner play have become common. They are known as 'joseki'. If you want to get any better at this game, you have to learn about joseki.
It is a good habit to learn a few joseki so that you aren't losing the game in the first 20 moves. ( There is a lot of debate about exactly
how you should learn joseki. Some say memorize them, some say don't. You don't have to worry about that yet. ) For right now, get a copy of '38 Basic Joseki' by Davies ( ISBN 4-906574-11-4 ) or some other beginner's joseki book and try to use those patterns in your games.
For a really basic introduction, while you are waiting for the mailman to deliver your book, I offer the following
micro-introduction to joseki:
PART 1:1) When your opponent has a 4-4 corner stone like this:
- Click Here To Show Diagram Code
[go]$$c
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . . W . . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$|-----------------------[/go]
...most of the time you will challenge his stone by playing

like this:
- Click Here To Show Diagram Code
[go]$$Bc
$$| . . . . . . . . . .
$$| . . a . . . . . . .
$$| . . . . . . . . . .
$$| . . . W . b c . . .
$$| . . . . . 1 d . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$|-----------------------[/go]
You can also play at 'a', for it is the same move, just a mirror image. Under special circumstances, you might play 'b', 'c', or 'd', but I recommend avoiding them until you know more.
1) When your opponent has a 3-4 corner stone like this:
- Click Here To Show Diagram Code
[go]$$c
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . W , . . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$|-----------------------[/go]
...most of the time you will do best to challenge it like this:
- Click Here To Show Diagram Code
[go]$$c
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . W , 1 . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$|-----------------------[/go]
...or this:
- Click Here To Show Diagram Code
[go]$$c
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$| . . W , . . . . . .
$$| . . . . 1 . . . . .
$$| . . . . . . . . . .
$$| . . . . . . . . . .
$$|-----------------------[/go]
PART 2:When he is first in the corner, and you are second, he has the advantage. So, generally, you will not get the corner. ( Or if you do, there may be some disadvantage to it. For right now, don't worry about those situations ) Usually, you will get a small portion - less than 50% - of the corner, or you will get pushed back along the side.
Those patterns often will look something like this:
- Click Here To Show Diagram Code
[go]$$c A common simple joseki when approaching the 4-4
$$| . . . . . . . . . . .
$$| . . 2 . . . . . . . .
$$| . . . . . . . . . . .
$$| . . . W . . . . . . .
$$| . . 4 . . 1 . . . 5 .
$$| . . . 3 . . . . . . .
$$| . . . . . . . . . . .
$$|-----------------------[/go]
or like this:
- Click Here To Show Diagram Code
[go]$$c A common simple joseki when approaching the 3-4
$$| . . . . . . . . . . .
$$| . . 6 . . . . . . . .
$$| . . . . . . . . . . .
$$| . . W , 1 5 . . . 7 .
$$| . . . 4 2 3 . . . . .
$$| . . . . . . . . . . .
$$| . . . . . . . . . . .
$$|-----------------------[/go]
WARNING: This micro-introduction to joseki is extremely simplistic. It glossses over many important ideas, and ignores many common exceptions. But it may get you through the first move with some idea of what you are trying to accomplish with your second and third moves.