The question of choosing a Linux distribution gets asked a lot Here are some views to help the newcomers.

Gentoo

Pros.

*A very effective system with a rolling release model.
*You can tweak Gentoo to become almost anything you like.
*Good software selection opportunities.
*You get what you build.
*Good for older computers as well.

Cons.

*It is hard for the newcomers because it is a source code based (everything is built from source by the user).
*No gui-installer: installation is done via terminal, which can get tricky.
*Things do not just work out of the box. They work when you have compiled them correctly. Learning Gentoo takes time.

Arch-Linux

Pros.

*A fast distribution with a rolling release model.
*Very good software selection via official repositories and AUR (Arch user repositories).
*A very robust system, which aims to keep things simple and effective.
*Arch has an excellent documentation and a great wiki.
*Good if you have some older computers.

Cons.

*No gui-installer – has a good documentation how to do it though.
*Packages can sometimes break and make the system a bit unstable: usually these problems are fixed quickly.

Option: Archbang

A distribution, which comes with Openbox desktop and has a gui-installer. Archbang is derived from Arch and gives a good change to test out and get started with Arch quickly.

Ubuntu (Based on Debian)

Pros.

*Good amount of programs available in repositories.
*Makes installing proprietary drivers very easy via synaptic and the multiverse repository.
*User can choose from bleeding edge up-to-date (release every six months) or LTS (long-term supported) versions.
*Has many official variations: Ubuntu (Unity), Kubuntu (KDE), Xubuntu (Xfce) and plenty of unofficial ones to choose from.

Cons.

*Ubuntu can contain bugs and crash quite easily if you always update the system as soon as you can get the newer version.
*Some people do hate the default Ubuntu choices, like Unity, as they are something that require a bit getting used to (for a majority of users).
*Unity along with other Ubuntu default programs might not be suitable for older computers as they require a bit more power from the hardware than older machines might be able to give.

Debian (Stable)

Pros.

*Rock solid and stable.
*Very good software selection to begin with.
*Very secure and focuses on stability and security.
*Good for older machines as well.

Cons.

*The software tends to get a bit old when time passes.
*Sometimes metapackages install too much stuff and quite literally do not keep it minimum, which can become an issue when computing with a low hard-disk space.
*Installing proprietary drivers can form a headache for new users.

Conclusions.

There are wide-variety of distributions out there and instead of giving a direct recommendation I would say test them out. Many distributions come with live-cd options, which gives you  opportunities to try them out before you install them.