On POSIX systems, termination signals usually have the following order (according to many MAN pages and the POSIX Spec):
SIGTERM - politely ask a process to terminate. It shall terminate gracefully, cleaning up all resources (files, sockets, child processes, etc.), deleting temporary files and so on.
SIGQUIT - more forceful request. It shall terminate ungraceful, still cleaning up resources that absolutely need cleanup, but maybe not delete temporary files, maybe write debug information somewhere; on some system also a core dump will be written (regardless if the signal is caught by the app or not).
SIGKILL - most forceful request. The process is not even asked to do anything, but the system will clean up the process, whether it like that or not. Most likely a core dump is written.
How does SIGINT fit into that picture? A CLI process is usually terminated by SIGINT when the user hits CRTL+C, however a background process can also be terminated by SIGINT using KILL utility. What I cannot see in the specs or the header files is if SIGINT is more or less forceful than SIGTERM or if there is any difference between SIGINT and SIGTERM at all.
UPDATE:
The best description of termination signals I found so far is in the GNU LibC Documentation. It explains very well that there is an intended difference between SIGTERM and SIGQUIT.
It says about SIGTERM:
It is the normal way to politely ask a program to terminate.
And it says about SIGQUIT:
[...] and produces a core dump when it terminates the process, just like a program error signal. You can think of this as a program error condition “detected” by the user. [...] Certain kinds of cleanups are best omitted in handling SIGQUIT. For example, if the program creates temporary files, it should handle the other termination requests by deleting the temporary files. But it is better for SIGQUIT not to delete them, so that the user can examine them in conjunction with the core dump.
And SIGHUP is also explained well enough. SIGHUP is not really a termination signal, it just means the "connection" to the user has been lost, so the app cannot expect the user to read any further output (e.g. stdout/stderr output) and there is no input to expect from the user any longer. For most apps that mean they better quit. In theory an app could also decide that it goes into daemon mode when a SIGHUP is received and now runs as a background process, writing output to a configured log file. For most daemons already running in the background, SIGHUP usually means that they shall reexamine their configuration files, so you send it to background processes after editing config files.
However there is no useful explanation of SIGINT on this page, other than that it is sent by CRTL+C. Is there any reason why one would handle SIGINT in a different way than SIGTERM? If so what reason would this be and how would the handling be different?
SIGTERM and SIGKILL are intended for general purpose "terminate this process" requests. SIGTERM (by default) and SIGKILL (always) will cause process termination. SIGTERM may be caught by the process (e.g. so that it can do its own cleanup if it wants to), or even ignored completely; but SIGKILL cannot be caught or ignored.
SIGINT and SIGQUIT are intended specifically for requests from the terminal: particular input characters can be assigned to generate these signals (depending on the terminal control settings). The default action for SIGINT is the same sort of process termination as the default action for SIGTERM and the unchangeable action for SIGKILL; the default action for SIGQUIT is also process termination, but additional implementation-defined actions may occur, such as the generation of a core dump. Either can be caught or ignored by the process if required.
SIGHUP, as you say, is intended to indicate that the terminal connection has been lost, rather than to be a termination signal as such. But, again, the default action for SIGHUP (if the process does not catch or ignore it) is to terminate the process in the same way as SIGTERM etc.
There is a table in the POSIX definitions for signal.h which lists the various signals and their default actions and purposes, and the General Terminal Interface chapter includes a lot more detail on the terminal-related signals.