To cross the border into the US you will require one of the approved Western Hemisphere Travel Initiative identifications. Currently these include, for a Canadian Citizen, the following:
A Valid Passport - valid for travel by land, sea, or air.
An Enhanced Drivers License or Enhanced Provincial ID which are currently issued by British Columbia, Manitoba, Ontario and Quebec. As the previous answers have stated, Alberta does not issue this form of ID and does not have any plans to start issuing it. They are valid for travel to the US by land or sea.
A Nexus Card - valid for travel by land, sea or by air to and from certain airports, this requires you to be interviewed and pre-cleared by both Canadian and US authorities. Although you can get one without a passport it takes longer than either of the prior forms of ID. It was designed for frequent travellers.
A youth, 15 years of age and younger, can travel by land or sea using an official birth certificate.
So it is possible to travel by car without a passport if you have one of the other approved forms of ID or are 15 and younger. However for an adult living in Alberta your most viable option is to get a Canadian Passport.