What is an antenna and what is the difference between an omni antenna and a panel antenna?

"What is an antenna": the antenna is a device that couples electromagnetic fields between free space and a circuit (radio, cell phone, etc.).

An omni or omni-directional antennais one that works equally well in all directions. Real antennas can only approximate this to a greater or lesser degree. In most applications, but not all, when someone says "omni antenna" they mean it works the same as you go around the antenna in azimuth.

The shape of a "typical" simple donut explains this very nicely. Going around the round edge of the donut, it is the same in all directions. But if you go up from the edge to the top of the donut, its shape changes. It's the same with a simple omni antenna. If you have one on your desk as a wireless access point and you walk around the antenna at the same distance, it will work just the same at every point in your walk. If you climbed up a ladder, though, then as you go up the ladder you will be going towards the top of the donut and how the antenna works will change.

A panel antenna is made very differently from an omni-directional antenna. It works a bit like a flash light, focusing all the energy in one direction. If you had a flash light sticking out from the front of a panel antenna, you'd see it stick out a lot in the front and only a little to the sides and up/down. Likewise, the panel antenna works mostly straight in front and only a little to the sides and up/down.