Is there any website / tool that allows me to see how many links I have been receiving over a period of time? I want to know if my site has attracted more links this month that last month, for example.
There is no tool that is completely accurate for what you're looking for, but check out SEOMoz. They have some good tools, and one of theirs does a decent job of what you're looking for.
There are a few out there...just do a Google search for "Backlink tool", you will get several tools you can try out.
With all those tools, you could never understand the total link popularity of your site. These tools have problems in picking all the Links. SEOMOZ is the closest I had.