'Packet editing' is using (altered) packets to trick the server into doing things it shouldn't, or couldn't, normally. Most of the time, you will sniff for packets using a Packet Sniffer application first. After that you can do many things with these packets, among which: Making Packet Filters, Botting or simply sending the (altered) packets directly to the server.
Packet Filters are as the name states, filters. These filters will look through the incoming/outgoing packets and see which ones match the filter's. The filter will then alter the packet like you told it to. Filters can block packets, change packets, reroute packets and more.
Botting is automating actions. This way you can make some programs work while not even being present at your computer. Packets often contain these actions (when you're using an online service, that is) so sending these packets using a program is considered botting: you're doing actions without actually doing these actions on your computer.
There are many user-friendly packet editors out there, eg: Winsock Packet Editor Pro("WPE"), Redox Packet Editor ("RPE"),... Each packet editor will have its own functions and ways of handling packets, but it will always be rather similar. See which one suits you and your level of expertise best, try out many.
Tutorials for packet editing, and usage of packet editors can also be found all over the web.
*more to be added*