When I send packets to another computer (both computers have wi-fi) is the AP required to communicate or can the other machine receive the information directly from my computer without the AP?
Since you mention two computers and an AP, I assume you are not talking about ad-hoc or direct connect, rather are referring to infrastructure mode operations.
In this case, all communication is between each station and the AP. This is because the station associates to the AP, establishing a relationship between the two devices, but there is no relationship between the stations.
This has a number of implications, but I will limit myself to two of the larger considerations in wireless.
First, just because a station can transmit to/from the AP, does not mean that any two stations associated to the same AP will be able to transmit to/from each other. To simplify, if you picture the AP's coverage zone as a clock (analog with hands) with the AP in the middle, a station at "12 o'clock" that is talking to the AP may not be able to hear a station at "6 o'clock" on the circle.
Second, the encryption key for security (whether based on personal/PSK or enterprise) is negotiated between the station and the AP. Even if a station can hear a second station, it won't be able to understand any of the encrypted data in the frame. (Note: in some cases you can intentionally decrypt this data, even if not the station and AP involved, however this is not normal operation.)
In ad-hoc networks wireless clients talk directly to each other without any intermediate device.
In infrastructure networks the APs are working as network bridges and all communication between wireless clients is going through them (even communication between clients associated to the same AP) This allows for advanced functionality such as client isolation.