The reference period for many (but not all) Census questions was Census Day, and this differed from decade to decade. The Census Bureau has an FAQ page on the timing of Census Day. You can see exactly how the question was asked in each year (for most years) on the questionnaire text page for AGE. For more details, I would advise looking at the original Census instructions and questionnaire forms available on the Census Bureau History website.
In some years (including 2000 and 2010) respondents were instructed to give the person’s age on April 1. In 1990 it was the person’s age at their last birthday. Other years collected the month and year of birth, but this is not available in the public use microdata for most years. I’ll also point you to this previous forum post on BIRTHQTR.