This has been covered in numerous threads, but I'll repeat it here: there are several ways to gain root privileges when they are needed (in this case, to run "make install" in the end if it works that way). They are mostly the same, but do have some differences. Some examples:
User becomes root, but the user's environment remains. In some distributions (Fedora is one, I think) some of the needed programs may be in sbin/ directories which are listed in the $PATH environment variable of root, but not in that of users. This means: if one uses "su" to become root, those programs are not
found just by issuing their name; instead full path to the executables needs to be written, or then the appropriate directories need to be added to $PATH before going on.
The same as above, but now the environment will not be what it was when the user issued "su", but will become that of the root user (as if root had just logged in). Now for example programs in directories listed in $PATH of root will be found, but on the other hand if the regular user (that ran "su") had some special environment variables set, they're not present.
Runs 'programname' with root privileges, requires sudo to be on the system and configured to allow this user to run 'programname' with it. Requires user password. If you have sudo configured (like on Ubuntu out of the box), do use it.
Then you can actually login as root, but that's about the same as if you just used "su -" to become root. It is also possible that when you become root using the "su" program, your current working directory is automatically changed to root home directory, which means you may have to re-navigate to the source directory..so if you're not sure,
and see that it's all right.
Typically you should run all the other installation steps (if not adviced otherwise) as a regular user except for the (usually) last one(s), where the compiled files are actually copied to your filesystem; you don't need root privileges here unless the files are copied to such directories where you don't have write permission. Most of the time, though, the "install" step copies something to bin/ and so on, and this does require you to have higher privileges, so practically you'll have to have root privileges then..but like stratotak
pointed out, you should run the earlier steps (configure script, make, ...) as a regular user so any files that may be created are also owned by you. If they're not, it's ok except that you usually can't move/remove them because you don't own them..
Reading the README/INSTALL files is good also because they'll mention if you have to set some extra environment variables during the compilation (not common, but possible) and if you should be doing some steps with certain privileges; for example if the installation failed unless you ran some parts as regular user, then some parts as root. There may also be information on how you should run the compiled software after it's installed; some programs don't run as root at all, usually for security reasons.