Merriam-Webster's dictionary defines a coward as:
"one who show disgraceful fear or timidity"
Opinions on what makes someone a coward may vary depending on who you ask. For some, a coward may be anyone who is unwilling to stand up for what he believes is right. For others, a coward is any person that runs from an altercation; too meek to stand up for themselves.
In Act II, Cesar has no desire to stay home even after he has received numerous warnings about this day. He tells his wife:
"Cowards die many times befor their deaths; the valiant never taste of death but once."
Of course a person can't die more than once, he means this metaphorically. In Cesar's time, a man who was labeled a coward might have died a social death. His friends and family would have lost respect for him. If he were a man of great wealth or power, his social standing may have suffered. Who can reverence a man who is too timid to stand up for himself?
On the other hand, a man who dies fighting, is a man who has died honorably, with his head held high. Mexican Revolutionary, Emiliano Zapata Salazar said it best when he said:
"I would rather die on my feet than live on my knees."
Cesar preferred the latter.