BAROTROPIC- Region of uniform temperature distribution; A lack of fronts. A perfect example of a barotropic environment is the southeast U.S. in the summer or the tropics. Everyday being about the same (hot and humid with no cold fronts to cool things off) would be a barotropic type atmosphere. Part of the word barotropic is tropic. The tropical latitudes are barotropic. There are no fronts in the tropics.
BAROCLINIC- Distinct air mass regions exist. Fronts separate warmer from colder air. In a synoptic scale baroclinic environment you will find the polar jet in the vicinity, troughs of low pressure (mid-latitude cyclones) and frontal boundaries. There are clear density gradients in a baroclinic environment caused by the fronts. Any time you are near a mid-latitude cyclone you are in a baroclinic environment. Part of the word baroclinic is clinic. If the atmosphere is out of balance, it is baroclinic, just as if a person felt out of balance they would need to go to a clinic.
You may run across the term quasi-barotropic- This means the fronts in the region are existent but weak. Often weak cool fronts will move into the southeast U.S. in Summer. Since the circulations created by weak fronts are weak, the atmosphere is not as dynamically unstable as it is in the case when the atmosphere is baroclinic. In most cases the atmosphere has some sort of minor temperature gradient in the troposphere outside of baroclinic regions.