We connect with people (and other living things) through the eyes, so the eyes are the most important part of just about any portrait that you do. So, here's a few quick tips on getting better portraits by paying close attention to the eyes:
1. Make sure they are in focus. If you use autofocus, focus on the eyes. If you are focusing manually, then concentrate on getting them in focus.
2. Have your subject look into the camera, it creates a feeling of connection.
3. Shoot at eye level. This is one of the more well known "rules" of photography -if you're taking a photo of a toddler or a pet, sit or kneel on the floor to get on their level. If you have eye contact, though, I find that this rule can be broken effectively. Note that shooting up at a person or pet can make them look more powerful or important while shooting down at them makes them look smaller. (I find shooting down at people/pets more effective when the angle is more extreme -shooting straight down at a toddler or pet looking straight up, for example.) But shooting at eye level is always the safest.
4. Pay attention to the light. If you can get your subject at the right angle to the light their eyes will light up and the color in them will be greatly enhanced. Pay attention to the light and take advantage of it when you pose people, but also look for it when you're taking candids.
5. Make a connection. A tiny difference in the eyes can make the difference between a blank stare, sadness, happiness, anticipation, fear, etc. Generally when people hear "say cheese" they turn to the camera and give a cheesy smile with dread, embarrassment, fear, or no emotion in the eyes. You'll find genuine smiles (and the real emotion in the eyes that follow a real smile) will really light up your photos, so try to make a connection with your subject -if it's a person, try to talk a genuine smile out of them. For pets, you can often get anticipation in their eyes with a favorite toy or treat held up near the lens to get their attention.