When guests arrive at your door, you might say "Welcome!" It means you are happy to see them, and hope they are about to have a good time with you. When they leave, you might say "Farewell!" It means you were happy to see them, and hope they are about to have a good time on their trip home. "Welcome" and "Farewell" are not opposites, but they are a pair like "Hello" and "Goodbye."
"Welcome/Farewell" are more formal and stronger than "Hello/Goodbye." You say "welcome" when people are coming to visit you in your house, and you are offering them "hospitality." They are coming to a party, or to dinner, or possibly saying overnight with you. Both might be the start of a little speech.
"Welcome! Come on in! Sit down, make yourself at home, it's great to see you!"
"Farewell! Have a good trip! It's been wonderful having you, don't be a stranger, come back soon!"
"Welcome" can be an adjective. In this usage, the opposite is "unwelcome." "Whenever I visited, my aunt always made me feel welcome. But my uncle was cold and distant. He never said anything, but somehow he made me feel unwelcome."