The claim is that 11 out of the 12 footballs provided by New England were deflated AFTER they had been checked by the NFL officials.
A few hours before the game starts, the officials check the footballs provided by both teams to make sure they are properly inflated. (Proper inflation is between 12.5 and 13.5 PSI.)
The footballs are then held by the officials until prior to the game, where they are handed over to the equipment managers for each team to take to that team's sideline area.
Now, at halftime, the officials checked the pressure on the Patriot's footballs again. (This is not standard procedure, as in this is not done in every game.) 11 out of the 12 footballs were found to be under the required minimum pressure of 12.5 PSI. Depending on which report/news article you read, they were under-inflated by 1 to 2 PSI.
Weather conditions (lower temperature on the field compared to the room where the initial pressure check occurred) could lower the pressure of the footballs, but it apparently did not do so (at least to the same extend) for the Colts' footballs. Hence, the charge that someone (or more than one) in the Patriots' organization deliberately deflated the footballs.