Mobile website testing is different from regular websites. User of mobiles are now want to have websites that are very easy to use and read. You never know, what user is using what mobile and what is browser of his/her mobile.
To address every thing is not easy how ever website under test can be put under certain check list. Some of the used item/images etc would be similar to the regular websites but many other things would be different.
Mobile websites are usually tested on different devices if available. In Mobile devices its really easy to test as it would a real test bed. But it would not be possible to arrange all devices, so we go for simulators.
Most of the simulators are easily available for downloads. You can download and install them on your machine. How every some of them would not be available for downloads and you would testing them online.
But some times client specify them for some particular mobiles. In this case it becomes a little bit easy.
You must be very careful, when you see website working very fine in one device but not the other. You report some bugs, now when these bugs are resolved, You must check the previous devices as well.