Golfers qualify for entry into tour events any number of ways, but most commonly by earning "fully exempt" status through money list rankings. If a golfer has not automatically qualified to play in a tournament, he or she can try to gain entry by playing in Monday qualifying.
It should be noted that not every professional tour event holds Monday qualifying; and the rules for Monday qualifiers - how to enter, how much it costs to enter, who is allowed to play - differ from tour to tour. (If you are interested in playing in a Monday qualifier, contact the tour that is staging the tournament for information.)
Examples from Two Tours
Let's give two examples of Monday qualifying on two different top-level tours:
LPGA Tour: Say the LPGA Anytown Open begins on Thursday. On the Monday prior to the first round, the LPGA holds Monday qualifying. All female professionals (including LPGA Class A or B teaching or club professionals) - along with two amateurs (with handicap indexes of 3.4 or less) selected by the tournament director - are eligible to enter Monday qualifying. But the field is capped at 60, and priority is given to those with LPGA or Symetra Tour status, then LPGA T&CP members, then other pros. Those golfers who enter play 18 holes, and only the top two finishers earn places in the field for the Anytown Open that begins on Thursday.
PGA Tour: On the PGA Tour, golfers with PGA Tour or Nationwide Tour status are eligible to play in Monday qualifying, and only the top four finishers advance into the PGA Tour event for which the qualifier is being played. But on the PGA Tour, there is also a pre-qualifier, open to all golfers, played prior to the Monday qualifier, and that awards places in the field for the Monday qualifier. So if you are a golfer who wants to take a shot and you meet the requirements (pro, or amateur with a course handicap of 2.0 or lower), you can enter the pre-qualifier. And if you do well enough there, you will advance into Monday qualifying, where you'll go up against PGA Tour and Nationwide Tour players.
The entry fee for Monday qualifying is usually around $400 or $500 for top tours. Again, for the specific rules of a tour's Monday qualifying process, including eligibility requirements and costs, golfers should contact the tour.