To wave is to move to and fro, like when you wave your hand. Hello there! To waive, with a sneaky "i," is to give up your right to do something.
Waves come in many forms. You can wave your hand, a hanky, or even a baseball hat to say hello or acknowledge someone, as in this example:
Fans around the green gave him a standing ovation, and Guan waved his baseball cap in acknowledgment.
A wave is also that big moving hill of water you can surf on:
Surfing students exercise on their boards before hitting the waves.
Other things are called waves, too — if a lot of people come in somewhere, for example, call it a wave of people. It's often used as a metaphor for things that act like ocean waves:
Maybe the great wave of social change has simply crested.
To waive, on the other hand, is to surrender, as in give up your rights to something. If you go scuba diving at a resort, you might have to waive your right to sue if something goes wrong. Here are some examples:
It also announced a mandatory arbitration clause, forcing users to waive their rights to participate in a class action lawsuit except under very limited circumstances.
Meanwhile, following budget cuts, the LOC board members have waived their salaries.
Waive is always a transitive verb, so you have to waive something. The other kind of wave doesn't need an object — although you can wave your hat, you don't have to. You can just wave. If you waive your salary, even for a good cause, you can wave goodbye to your money!