To beware is to be cautious and on your guard. The riskier the situation, the more you need to beware.
A "Beware of the dog" sign means there's a dog in the house who doesn't care for visitors or new people. When you beware, you become very cautious. You look both ways and use extreme caution. This word can be remembered if you break it into two parts: to beware is the same as "to be wary." When you're wary, you're concerned and careful. People in dangerous jobs like the police and fire department need to beware every day.