If you do something awkwardly, you do it with very little grace, in a klutzy or bumbling way. When a first date goes awkwardly, you might be tempted to skip a second date.
People behave awkwardly when they're embarrassed or self-conscious — you might act awkwardly as you stand on a stage, accepting an unexpected award. You might also move awkwardly in a purely physical sense: "The cast on his leg meant that he shuffled awkwardly down the hall on crutches." The original meaning of the adjective awkward was "in the wrong direction," a combination of awk, "back-handed" or "turned the wrong way," and -ward, "toward."