Carol -
Be sure you understand the difference between "memory" and "hard drive capacity". The whole process will get hugely confusing if you don't.
"Memory" is the stuff that the processor uses while it is turned on and doing something. Memory is fast, but everything in it goes away when the computer is truned off. These days, 512MB or 1 GB of memory are good numbers - don't settle for 256MB, for instance. And 2 GB is probably overkill, though there is nothing wrong with it if you want to pay for that much.
The "hard drive" is where the computer stores everything when the computer is off. All of your programs, and all of your data ("data" is pictures, documents, and so forth) are stored on the hard drive. I think 40 GB of hard drive space should be plenty, UNLESS you are really into pictures, videos, and other graphics. Graphics is the big hard-drive hog - texty things (like your medical records) are relatively tiny. If you think you will keep thousands of pictures (.jpg files) on your computer, then you may want to go to 80GB or more. Similarly, if you think you will be storing movies or videos on your computer, you may want more.
Since laptops get carried around, they are subject to more mechanical shocks than home desktop computers. This means that their hard drives, in particular, are more subject to failure. I would strongly suggest that you have some method of backup for the important stuff on your hard drive. A portable hard drive has been suggested, and that is a good solution. USB flash drives, also known as "thumb drives", are a good solution for backups, and are quite inexpensive. You use a flash drive purely for backups, so the limitation of a couple thousand read/write cycles is not a big one. And of course, if you have two laptops, you can keep a copy of each important file on each computer - they back up each other.
By the way, you will certainly want a CD player on your machine. For a few more bucks, you can get a CD player/recorder (also called CD-RW, for read/write). This enables you to put your backups on a CD. Once the backup has been written, you can't change it - but it will never go away, either. When it is time to back up some more files, you just slide a new CD into the machine, and back up whatever has changed since the last backup. Recordable CDs are cheap and almost indestructible.
No matter which way you choose to go, remember that the big pain about backups is that you have to remember to do it. It is not automatic. And of course you have to choose what to back up. A quick snapshot of a flower may not be important - but your medical records certainly are.
One further note. When you choose your new computer, you will have to choose an "operating system". The choice is between Windows XP and Windows Vista. XP is 3-5 years old now, well-proven, well-understood, well-behaved, and well accepted. Vista is new, and requires lots of memory (1 GB is not enough). Beyond that, Vista is glitzy, but my understanding is that it is not significantly different functionally from XP. At this point in time, I would avoid Vista. In 2 or 3 years, when the bugs have been found and ironed out - maybe Vista will be good. But not yet.
By the way, Windows XP comes in two flavors, called "XP Home" and "XP Pro". XP Home is significantly less expensive, and perfectly adequate.
Hope this helps.
Bill