ok kk here's a starter for ten - feel free to ask questions if this doesn't make sense or you want more detail!
Until the last year or so the two digital camera types available were:
DSLRs - which pros/serious amateur photographers use.
Excellent results, flexibility to be more creative (e.g. play with depth of field to get blurry backgrounds), control over all elements of the photo (light, speed etc), you ca also buy numerous lenses and filters to get different effects (e.g. zooming to get distance wildlife shots). Downsides are that they're expensive and big and heavy plus you do need to know what you're doing to get the best out of them.
Compacts - this is the type you currently have and they're good for everyday snapshots. They're much cheaper than DSLRs and really range in quality depeding on what you spend. They're usually small and light and easy to use but you're very limited in terms of control over the finished photo.
And then more recently a new type has been developed which is a hybrid of the two above, it's called the CSC (compact system camera) and it's the type I use. It's much smaller than a DSLR but gives you similar control and results - the technology isn't yet good enough for pros but is great for serious amateur photographers. You can buy different lenses and filters just like with DSLRs. The downsides are that they're still bigger and heavier than a compact and they're expensive.
Your choice should probably be between a Compact and a CSC. If you go for the former then I've always found the higher end Sony Cybershots give good results. If you fancy a CSC then I'd go for either the Olympus Pen EPL1 (smallish and with a retro look, it's what i have) or the Panasonic Lumix (bigger and heavier).
It would also be worth downloading something like Google's Picasa as it's free and very quick and simple to use to fix red-eye, crop, make improvements to contrast etc and also email or file photos into albums.