I'm a web designer and I was just wondering what the restrictions are on these images. I would like to use them as a background image on a few sites. is this allowed?

No. All images are copyright their respective owners. Get permission from the artist first. That goes for all images you find on the net unless the images are specifically offered as Public Domain. That said I have found at least two members here with galleries consisting of at least 70% ripped images. Ripped means they post images they did not make them selves and have posted them here without permission in violation of copyright. This is a violation of the DesktopNexus terms of service and a crime. But that doesn't seem to be slowing them down any.

if it doesn't say copyright on the bottom of the picture
or on the page of the picture what is a person supposed to do?
guess? if i find pictures i like anywhere on the web and they are offered
to the public then they are up for grabs.

if i see or can find a copyright i don't use it for web designing.

The pictue that I want to use for a website does not have a Copyright or a C symbol on the picture does that mean that you should be able use it.

Desktopnexus cannot give permission for you to use any image found here. We do not own any of the wallpapers. While we ask our members to only use Public Domain images, many that are uploaded here are found free of any markings that show copyrights because they have already been cropped of any ownership and found all over the internet. This does not mean they are free to use! While it is not our responsibility to check every wallpaper that is uploaded to DN, we have a system in place just like all sites on the internet that accept public uploads, in compliance with DMCA laws to have an image removed quickly when a copyright is claimed.
Many of our members DO create their own wallpapers, so check with the uploader to see if it is their own work..
You might also check Tineye for an indication of other sources for that image.