Read old articles instead books. I'd like to know if there is a site, or maybe a collection of books, where I can read old articles in mathematics in order to study topics directly from the source, instead reading books in the field. I always feel that my study is incomplete when I don't know the motivation, or the history backgrounds, of the definitions or theorems (even when I read good and classical books).  
 A: Stephen Hawking's God Created the Integers: The Mathematical Breakthroughs That Changed Historycontain many old articles. (as Bruno mentioned already)
Timothy Gowers' The Princeton Companion to Mathematics contain concepts in article form by contemporary mathematicians.
Stewart Shapiro's The Oxford Handbook of Philosophy of Mathematics and Logic contains many papers on logical concepts, again, by contemporary authors.
Blackwell guides can be useful such as The Blackwell Guide to Philosophical Logic (Blackwell Philosophy Guides).
As for online versions please see the related link: List of Interesting Math Blogs 
And you are perhaps already familiar with Stanford Encyclopedia of Philosophy; for mathematical biography Mactutor History of Mathematics is resourceful.
Finally, I personally visit Interactive Mathematics Miscellany and Puzzles to understand basic concepts.
A: I haven't browsed their collection in detail, but Numdam has a lot of good stuff.
A: Many of the old journals are available online, though some may require subscription.
For example, Crelle's Journal, going back to 1826, is freely available at
http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=238618
Or you could consult the works of Gauss
http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN235957348
or Euler
http://www.eulerarchive.org/
A: Many authors have pre-publication versions of articles they wrote on their personal web page and sometimes old articles are now in the public domain. You can often locate these articles using either: http://scholar.google.com/  or  http://www.scirus.com/srsapp/ by searching using some appropriate string or using the author's name.
A: http://www.archive.org/ has lots of old stuff.
A: There's a wonderful and comprehensive site for finding old papers and book about mathematics.  It's called a "library".  It probably even has comfortable chairs for sitting and reading...
