Cygwin compiler is used to compile POSIX programs (like those used on Linux) to MS-Windows. It hardly ever useful on a Linux desktop, unless you want to "cross-compile" programs for MS-Windows from your Ubuntu desktop.
I suspect that you what you want to do is to download software that comes in
.tar.gz archives (also known as "tarballs") and use these on your Ubuntu desktop. To do that you need a compiler, but not Cygwin - you need GCC. Use Ubuntu's Software Center to install "gcc", then open your tarball (the archive that ends with
.tar.gz) to an empty directory, open a terminal and
cd to that directory then run
./configure and if that completes without errors, run
make install. That will get the software installed for your local user. You might also want to read the Compiling Easy How To document to learn how to best approach this process.
Its important to note that most software you'd want to use is already included in Ubuntu's Software Center, so I suggest you always start your software search there - software distributed by Ubuntu is much more likely to be safer and better then what you download from the internet. Even if you can't find exactly what you are looking for likely you can find useful alternatives.