A putative spouse is someone who you have a good faith belief that you are married to, but where the marriage is not legally valid.
Let's say for example, you go through the process of becoming married but for some reason your marriage certificate was never filed. Technically, the marriage is not legally valid, but you and your spouse thought that the marriage was valid. This is a situation where your spouse would be a putative spouse. Or, let's say for example, that you thought your prior divorce was complete (but it wasn't), and you later were legally married to someone else. Then your new marriage is not legally valid, and your new spouse is a "putative spouse".
The rights of a putative spouse can be tricky to determine. Ideally, if you find yourself in this situation, it should be dealt with before the death of either spouse. If there was a prior divorce that was not properly dissolved, it should be done so immediately. And, if your current marriage was defective because of some technicality, steps should be taken to make sure the marriage formalities are followed. Of course, this is often easier said than done, because oftentimes problems of this type never come to the forefront until after one of the spouses dies and someone from the past comes out of the woodwork.
Some court decisions have held that a "putative spouse" should be given the same rights as a legal spouse. Thus for example, he or she would be entitled to whatever a legal spouse would've been entitled to of your estate. However, other courts have taken a different approach when there was the presence of a surviving legal spouse, as well as a surviving putative spouse.
As a result, if you are in the process of getting married, be sure you get the proper documentation, including a validly filed marriage certificate, or if you're getting a divorce, make sure to obtain a final divorce decree from the court signed by the judge, so that there's no doubt in the future.