What happens if my husband and I move in with his elderly mother?

If she cannot pay for her own nursing home car, the state health system can go after the home the take it to pay her nursing home bills: people are required to use up all their assets, including real estate, before Medicaid kicks in, and if they don't, Medicaid can then recover those assets.
If there is a mortgage on the home (including a reverse mortgage) and payments are not made, the home can be foreclosed on.  If it is foreclosed on, you'll have to leave once the foreclosure process is complete.
If she passes away, any mortgage comes due, and must be paid, immediately, or else the home may be foreclosed.
If she passes and there is no mortgage, the home will go to whomever her will (if there is one) says it goes to; or if no will, it pass by the rules for intestate succession, which will presumably be to your husband and any siblings of his. Whomever gets the home can require you to leave, unless your husband is one of those inheriting (in which case he can't be forced out, but if there are other heirs, they could eventually force a sale of the home, with the equity being shared between the heirs).
It is not your home: you won't be able to stay there forever. But all of the things discussed above typically take months or longer to play out: if you are saving money to use to rent or buy another place in the future, you should have ample warning of the need to do so.