Typically landlords ask their current tenants whether they are renewing leases in the new year...you'll start to see lots of 'for rent' signs go up in Jan/Feb when the new semester starts. Most students begin looking then and as DannyV said, they sign when they find one they want.
That was my experience looking for houses around the McMaster area, I'm not too sure if it's different for appartments, but hopefully that helps a bit