Indoor palm trees are beautiful feature plants for your home. They'll give it a nice, chilled-out, tropical or desert feel. Some of the young palms are tiny and can make nice desk or table plants. The larger palms will stand nicely on the floor. They make very good office plants.
The palms I've written about below come from all around the world and it's difficult to generalise on their care. I suggest reading and following the specific care instructions for the palms you choose. Apart from the obvious watering and light considerations, pay attention to the humidity requirements.
Many palms are prone to root rot. Prevent this by ensuring good drainage and not letting water sit in the saucer under the container.
Kentia Palm (Howea forsteriana). Also called the Sentry or Thatch Leaf Palm. excellent, slow growing, very tropical in appearance. Place in a location receiving bright indirect light. Water the entire root bulb and don't allow it to dry out completely, nor overwater it. Let the top couple of inches of soil dry before watering.
The Lady Palm (Rhapis excelsa) is a fan palm . It grows from multiple stems and enjoys shade or partial shade. Water it well in summer, but make sure that it drains completely. In winter, reduce this to once every two weeks or even less.
The European fan palm (Chamaerops humilis) is originally from Spain. It's a fan variety of palm that grows well indoors. Put in medium levels of light and water well, letting the soil dry partially between waterings.
The Parlor palm (Chamaedorea elegans) is a small palm that works well inside. They prefer filtered light and watering of the entire soil base around the plant. Only water again when the top inches of soil are dry.
Pygmy date palm (Phoenix roebelenii) is a small palm with fronds coming from a small trunk. It likes bright, indirect light. Water with warm water to keep the soil moist.
Areca palms (Chrysalidocarpus lutescens) are very popular beginner palms being reasonably affordable. They enjoy medium to bright indirect light. Keep the soil moist but not wet.
Palms stand well on their own as feature plants, and they need a fair amount of space around them for their fronds to grow.