A manicure just trims your cuticles, shapes the nails and polishes. Sometimes they give you a hand massage. It doesn't last very long. Maybe 1-2 weeks.

Acrylic nails are plastic tips glued to your nails to make them longer. They then brush on liquid acrylic and shape the nails to look natural, and polish. They stay on a long time, but when they grow out you have to have them filled in. Usually in 2-3 weeks. Also, acrylic nails damage your real nails. They look nice, though, and are sturdy.

Gel nails are the same as acrylic, but they put another, smooth layer over the acrylic, which is "Set" under a UV light.
These are crystal clear and can be worn with or without polish. They are even stronger than acrylic nails, too. But they are usually more expensive and the fill-in take longer.

Hope this helps.