I'ld like very much learning to program under Linux. What better way to achieve that than creating your own application, no? The problem is I don't have any ideas...
However, there are some things I have to mention:
1) The programming language must be c/c++
2) I don't want anything network-related
3) I'ld like very much like my application to manage processes, memory or other low-level stuff ( I'm willing to learn as much as I can about all these)
I hope I was clear enough...So,ideas?

The best way to get started is to do something that you want. Is there something that annoys you? Do you wish some existing tool did something that it doesn't do now? If so, that's your itch - scratch it. If you're feeling generous, offer the fruits of your labor to the rest of the world. In that one step, you'll have become part of the solution.

I agree with macemoneta, but to offer a possible idea for learning, how about building a shell? In one of my courses at university the final project was to write a shell (in C) that could handle I/O redirection via pipes and the stream operators. In other words, it could do stuff like:

myshell $ cat myfile.txt | grep "linux" > output.txt

That project taught me a lot, especially in process management and text processing.

I usually start some sort of "useful" project that results in incidental components (such as parsers, list classes, containers, etc.) Rather than being "smart" and using pre-built or standard libs, I choose to write things myself from scratch. I learn by having to research solutions to problems I run into, and a whole lot of experimentation. This also teaches me how to write better APIs, manage memory, and write more reliable code. I then try to break my code by subjecting it to bizarre situations. In all reality, I never end up finishing the big project, but instead perfect the small incidental components that result. I post the better ones on SourceForge.
ta0kira

Thanks, you've been very helpful.