I would say that the "majors" of amateur golf are those that offer the winner a chance to play in a professional major:
U.S. Amateur, British Amateur, U.S. Mid-Amateur, U.S. Public Links
The winner gets into the Masters, and the U.S. Amateur winner also gets into the U.S. Open and the British Open. The British Amateur winner also gets into the British Open.
Then there are those established amateur events that get the strongest fields and are considered a level below the majors:
Western, Northeast, Sunnehanna, Porter Cup
Then there are many more a level or two below those:
Southern, Dogwood, North & South, Monroe, Azalea, Pacific Coast, Southwestern, Southeastern, Rice Planters, Dixie, Pacific Northwest, Trans-Miss, Eastern, Cardinal, Canadian, Sahalee, Stocker Cup, Middle Atlantic, Sahalee
And then:
Jones Cup, Labor Day, Palmetto, New Year's, Birmingham, Florida Azalea, Gasparilla, and many more...
And then there are those club invitationals that are held at the truly great courses:
Crump Cup, Hugh Wilson, Coleman, and many others
Some of the amateur events have qualifiers but for most the way to get in is to contact the tournament committee (go to amateurgolf.com for contact information or look for individual websites which many of the tournaments or organizations have) and submit your competitive playing resume and invitation request.
For many of the invitationals, though, a personal contact is necessary to gain entry into the field.
If you want to play high-level tournament golf in tournaments that are a little easier than the others to get a shot at (either by invite or qualifying, though you'd likely have to travel to attempt to qualify) at great venues, you can't go wrong with:
Trans-Miss Mid-Am at Prairie Dunes
North & South Am, Pinehurst #2 (and #8)
Azalea Invitational at CC of Charleston, SC
Palmetto Am at Palmetto GC, Aiken, SC
Holston Hills Invitational at Holston Hills CC, Knoxville, TN
amateurgolf.com 2-Man Championship at Bandon Dunes
Of these, the North & South and Azalea are the toughest "nuts to crack" but do have local qualifiers.
Throw in the U.S. Am at Merion and the British Mid-Am at Muirfield, and there will be plenty of good venues for amateur golf in 2005!