A virus is computer program that writes itself to another program, which would then write itself to other programs and so on. In Ubuntu, only root is allowed to write to executables. This means that a virus would have to gain root access before it could infect your system. So even if the virus did reach your computer, it would be very difficult for it to actually infect it. It is not impossible, but difficult.
But a virus is not a virus unless it is able to actually spread. It is very uncommon for users to copy software directly between themselves. Instead, you'd download the software from the distributor, who has compiled the code from source and then signed it. First of all, this means that a virus would be incapable of spreading because it isn't being copied. But it also means that Ubuntu would actually refuse to install infected software quite simply because the certificate would no longer be valid. When you install Ubuntu from a CD, you can test the CD for errors. It's the same principle. If the software has been changed, then it is no longer valid. Whether that's because of tampering or a bad download, is actually quite irrelevant.
In other words, there are many reasons why viruses will have an insanely small chance of surviving in Ubuntu. So much that it can be considered almost impossible.
However, viruses are only one kind of malicious software. The anti-virus manufacturers like to use that expression for everything because of the fear factor. It is completely possible to make other kinds of malware, like trojans. Actually, this is a little bit easier on Ubuntu than other operating systems, using PPAs for instance, but also easier to detect. So in other words, even if Ubuntu is mostly immune to any kind of virus, it is not safe to assume that it is safe to run just any software.