"Can this be fixed or do I have to buy a new computer?"
When you say a computer, it includes a lot of stuff, majorly classified into the following:
The hardware, or the physical parts of the computer. Those stuff that is inside the "CPU" box. Keyboard, mouse and monitor are also part of the "hardware" but since they donot directly impact the "working" of the computer, we will not go into them.
The software, the programs that run on your computer. This includes the operating system (Win XP in your case), the internet browser (Internet Explorer, Firefox,...), Microsoft Word, Notepad etc...
The operating system needs some "minimum configuration" of hardware on which it can work smoothly. Once it is working, as long as the physical parts are in working condition, it is most likely that you need not replace them.
So, if you were able to work smoothly sometime before with this computer, and it is getting bumpy off late, you need not "buy a new computer" (in most cases).
What would have happened is that your computer got "loaded" with software over a period of time, and this made it slow.
The best solution is to find someone who has some knowledge of the "computers" and is willing to help, so that he/she can clean some of the stuff that might cause the slowdown
If you have a CD of the operating system (XP) try to format the system reinstall it. Reinstalling a operating system means that you lose all that is on your comp, those pictures, movies... so you might want to "back it up" (on CDs, own or borrowed external hard disk) so that you can "put them back" after the reinstall.
If you are not comfortable with computers, cleaning up stuff, dont try reinstalling the operating system yourself. Try to get some help.
To answer your question: very high possibility that you would not need to "buy a new computer". In the worst case you might have to change some parts (memory, network card...).
Also, as a sidenote: be careful on what you download from the internet, or what sites you visit. It is a scary world out there.