Free-agency works similar to real life free agency. There is a set period when free-agency begins and when the spring training begins. The majority of free-agents want to sign during this period.
Free-agents typically receive multiple offers. Early in the FA period, they will receive and weigh many offers. As the time gets closer to the end, they will be less patient. Each player is different and has a different time table. Some will sign sooner than others.
Each player has a different agent, too. While you'll typically get a chance to adjust your offer after another owner outbids you, it's not always the case as some players will have run out of patience or simply gotten an offer they really wanted. They will then sign on the spot rather than risk going a few more rounds.