Become a musician instead of simply being a singer. Just as much as a piccolo or clarinet, the voice is an instrument, and a cappella singing requires you to play that instrument well. Take voice lessons from singers who know more than you do and sing scales and perform exercises every day. Warm up your voice each time you sing.
Perform without amplification when you can. While you'll likely sing into a microphone while onstage, listening to your own unamplified voice, or those of your group members, will help you with the intangibles of singing a cappella: figuring out how to fit your voice to others or to the acoustics of the room.
Plug your ear with your index finger while you are rehearsing a particularly difficult passage. Not only will this allow you to use your head as a resonance chamber, ensuring you'll hear the sound you're really making, but you'll also shut out any noise that will distract you from your part.
Listen to the songs being sung without your part if you're in a group. Rotate every few bars as all of you sing the song so each of you can hear the a cappella harmonies minus your note.
Fill out the aural experience with sound made in other ways. You can add percussion by hitting different parts of your body in different ways. Hitting the broad part of your chest with an open palm will create a higher-pitched slap; a closed fist in the same place can mimic the low-pitched thump of a bass drum.
Allow the rest of the a cappella group to shine as well. According to Lovetosing.info, "let go of your ego and sing for the betterment of the group--trying to achieve higher and higher levels of harmony." If you're singing alone, sing in such a way that serves the song. Even if you are a good a cappella belter, a sad section of a song might demand a quieter, calmer tone.