Each league has a pool of free agents that includes people cut from other teams and players skipped in the amateur draft. Some general guidelines on the free agency system:
You can only bid on free agents in a league which you own a team in, and the team in that league is the one that's bidding on the free agent.
Free agents will only sign during the preseason and regular season. They will not sign after the last day of the regular season.
Free agents consider offers during the league's simulation; they do not make decisions after each contract offer is submitted.
Free agents will wait a minimum time before making a decision (to give other owners a chance to make counteroffers). The actual time depends on the sim rate.
2 games per day leagues: minimum 4 days from first offer, one day from most recent offer, maximum 8 days from oldest offer.
4 games per day leagues: minimum 2 days from first offer, 12 hours from most recent offer, maximum 4 days from oldest offer.
8 games per day leagues: minimum 1 day from first offer, 6 hours from most recent offer, maximum 2 days from oldest offer.
Free agents will always make a contract decision if the highest offer is at least double his base (minimum asking) salary.
Note that "days" are calendar days, not sim days.
When you submit an offer to a free agent, you have the option of offering more than their asking salary. (This is common when multiple teams are bidding on the same player.) If you sign a player at more than his asking salary, the amount offered over the player's base salary will be reduced by 50% each year. If the amount offered ever gets to 10% or less of a player's base salary, it is dropped altogether.
Free agents will not make a decision solely based on salary if he has multiple contract offers. Some of the factors which influence a player's decision to sign with your team include:
The winning percentage of your team. (Players prefer winners.)
Whether or not the player is better than the best player at the same position on your team. (If he is better, he's more likely to start, so he's more likely to sign. If he's worse, he's less likely to sign, and may look to play elsewhere.
The overall payroll of your team. (Higher payroll makes a player less likely to sign, as there is a greater chance of him being cut in the future for salary cap reasons.)
Whether you ever cut the player from your team in the past. (Players don't like to go back to teams that previously cut them.)
When a free agent is signed, he will join either your AAA or low minors. If the player has no active roster experience and you have room in your low minors roster (low minors has a maximum of 10 players), he will be assigned there. If the player has one or more years experience, or if you have insufficient room in your low minors roster, he will be assigned to AAA.