Making work fun promotes teamwork and builds productivity. Humor can also diffuse a tense situation, as long as the humor is not at someone else's expense. Gossiping can be hurtful to others and can irrevocably damage your working relationships with them. Never say anything about anyone that you would not say to his or her face. Likewise, take responsibility for your actions - even your mistakes.
Accountability is another important element of trust.
Occasionally even your best efforts may not produce the desired result. In these instances, you may need to take further steps to mend the situation. Approach the individual for advice or ask him or her to join you for coffee. Sometimes just getting to know someone and gaining a better sense of his or her motivations can improve the relationship. Ask if you've done something to offend your coworker, and emphasize that it was not your intention.