I am creating an application related to files. And I was looking for ways to compute checksums for files. I want to know what's the best hashing method to calculate checksums of files md5 or SHA-1 or something else based on this criterias

The checksum should be unique. I know its theoretical but still I want the probablity of collisions to be very very small.
Can compare two files to be equal if there checksums are equal or not.
Speed(not very important, but still)

Please feel free to as elaborative as possible.
md5 tends to work great for checksums ... same with SHA-1 ... both have very small probability of collisions although I think SHA-1 has slightly smaller collision probability since it uses more bits
if you are really worried about it, you could use both checksums (one md5 and one sha1) the chance that both match and the files differ is infinitesimally small (still not 100% impossible but very very very unlikely) ... (this seems like bad form and by far the slowest solution)
typically (read: in every instance I have ever encountered) an MD5 OR an SHA1 match is sufficient to assume uniqueness 
there is no way to 100% guarantee uniqueness short of byte by byte comparisson
File Checksums in Python