Packet usually refers to AX.25 which is an amateur version of the X.25 data protocol. In AX.25, data is assembled into "packets" of a certain length. The transmitting station calculates a checksum that is transmitted along with the packet. The receiving station calculates a checksum on the received data and if the calculated checksum matches the transmitted checksum, the receiving station transmits an acknowledgement (ACK) indicating that the packet was received correctly. If the transmitting station receives an ACK, it will transmit the next packet. If the transmitting station doesn't receive an ACK, it will retry sending the failed packet a certain number of times until it gives up. Packet can be used on HF at 300 baud, on VHF at 1200 baud, and on higher frequencies at higher baud rates.
I'm a little hazier about PACTOR, but I've gathered that it is a evolution of packet and AMTOR. AMTOR is similar to RTTY and is the amateur version of SITOR. The advantage of AMTOR over RTTY is that AMTOR has some error detection which RTTY does not have. PACTOR comes in four "flavors", PACTOR, PACTOR II, PACTOR III, and PACTOR IV. PACTOR improves the throughput of AMTOR by using some data compression and wider bandwidths.
Either packet or PACTOR require some sort of Terminal Node Controller. Traditionally, TNCs were hardware devices that assembled/disassembled the packets and did the modulation/demodulation of the audio signals that are used over the air. For packet, there are software TNCs that run in your computer so all you need is an audio interface between your radio and your computer. Note that PACTOR IV TNCs are only available from a single source and are rather expensive.
Winlink typically uses packet on VHF/UHF and some flavor of PACTOR, or Winmor, on HF. Winmor was developed by the Winlink team and does not require a TNC.
If your plan is to dabble with Winlink on HF, download and install Winlink Express and use Winmor. You won't need a TNC.