If the gaps are between the door and the door frame, you can buy self adhesive weather stripping at any home improvement center. The weather stripping can be placed on the door, or the door frame, and should help to close those gaps.
If the gaps are between the door frame and the wall, caulk is an effective solution. A tube of good caulk costs anywhere from $1.50 to $3.00 and a cheap caulking gun should be between $5 and $10. It is usually available in a couple basic colors and clear (goes on white, dries clear).
Good Luck!How do I seal my front door properly?apartment ?
untill the land lord/ess gets the right seals you can use duct tape.stick one edge on and make a fold deep enough to cover the gap then stick the other edge on.it will close with the door ,keep out weather . use shorter pices only if need be. not always easy to handle.How do I seal my front door properly?Until you landlord takes care of this for you (like he/she should), try using a rug to put against the crack. It's temporary, but it's also free! (assuming you have a rug)
