A bathtub with a door is just what it sounds like. It is a tub basin that you may lay down in and soak during a hot bath after a long day. A bathtub with a door means that any person who has to climb in and out of the bathing area is not going to risk being hurt when they do so. Rather, a bathtub with a door means a simple swing out of the door lets the bather shuffle right into the basin, rather than having to step over a large lip.
The design of a bathtub with a door will change based on who made it, but most are built so that the water is running toward a drain that is down a slope from the door. This means the bottom of the basin is angled so that water runs to the middle or to the end of the tub that is away from the door. It might mean there is not a totally even floor on the basin, but it is not so steep of an incline that you will have a shallow end and a deep end.
Rather, your bathtub with a door will be almost exactly like any other tub, except without the risky step in. This matters for an elderly person who cannot quite lift their legs like they used to, or for a person with a leg injury who has to take a bath and keep the cast out of water, or for a disabled person who cannot climb in and out of a normal tub. They also mean a lot of help for any person who is a care giver and needs to help a disabled or elderly patient with a rinse and wash. They will hold the door open, help the patient into the tub, then close the door and start the water as they make sure the patient does not fall.
A bathtub with a door will not cost much more than any usual tub. In fact, if you shop around you might find a great deal on a bathtub with a door that ends up costing less. They tend to have handles in the basin itself, as well as controls for the water flow and heat in a strategic place so the bather can reach them. Be sure to ask about delivery options too, because they may take extra care to install.