My first distro was Slackware...it seemed fitting as a new distro, but not nearly as friendly as the more popular distributions. It depends on what type of experience you want to have...and what type of computer user you are. You will more than likely not like the first distro you try to use...but then again, you may. I will rank the below is order of user friendliness in groups...
Redhat, Mandrake, Suse, etc are all friendly to a new user, but are not the best for a slow machine. These would be the distributions that have everything prepackaged for the whole linux experience...and more than you will ever use. These are in no way cut back for a slow computer, but should install well with almost any hardware.
I kind of put Ubuntu and Yopper in-between the top group and the next group. You get much more an open source feel from these distributions that are not as commercial as the above, but are still user friendly. Both are based off of Debian, and use a great package management system for installing. You may need some configuring here, but it might be the right group if you want to learn...though I would not suggest Yopper for a lower end computer. Ubuntu would be good, but you may need to move away from Gnome to a window manager that requires less resources. Ubuntu would be my main suggestion, and is what I use right now...I moved from source base distributions to Ubuntu...I love the package management, but hate being away from the customization, and can see myself going back to a source based distro on my desktop soon...it seems like I do not learn as much on a day to day basis about Linux using Ubuntu than I did using a source based, but more because it just works than anything.
Below that group are what I call the classic gnu/linux distros. These include Debian and Slackware. Both do not give you comforts such as graphical configuration tools...Debian does have a great package manager, but I have always had a harder time installing it than even the source based distributions. Slackware is pure...there is no fluff, although there are plenty of utilities, and no package management. It was my first distribution and a great one to learn on, but not a great one to try to update on a day to day basis...it is a "get it working, and don't touch it again" type of distribution to me. However, it is great for slow computers and servers. There are now package managers for slackware, but I haven't tried it since I made the initial move to gentoo.
Finally, the source based distros...these are not entirely friendly to new users, but you will learn more than you ever care to know about Linux from them. These include Gentoo, Sourcemage, and for an extreme, Linux from Scratch. They also do not work entirely well on slow computers...kind of. The problem is that you have to compile every program you run...everything runs faster if you do things right, and you can decide how much "fluff" you want, but it will always take you much longer to get a running system than a prebuilt distro, and will take you longer to get a new package than almost any other means. You will run into problems eventually...but hopefully you will learn to recover from them by research. After you run a source based distribution for a few years with the number of problems that I had, you will probably be ready for Linux + certification if you can retain everything you had to do to fix the problems.
My suggestion to any Linux newb that wants to learn Linux is always Ubuntu...it seems like a middle of the road distribution that is on the cutting edge and friendly, but still requires some config file editing every once in a while...although it is rare. For Business users that have no desire to learn or grow with Linux, I suggest the popular more commercial distributions like Redhat, Suse, etc...and the reason that I call them commercial is that that is who they seem to cater to, businesses...not that they are any less free than other distributions (except for Enterprise editions/etc.)
There is a section on the linuxquestions wiki for choosing a distro, as well as a forum for distro reviews. You might also check distrowatch that is a great listing of distributions...(check google for the address, I cannot post links.)
It can be overwhelming...my advice is to just pick one and stay in some linux...I had to stay in it for two months without ever going back to Windows to start to understand it, and honestly it was Gentoo that drew my attention. AFter that I started to love it SO much more, I started to enjoy computing again. Good luck choosing!