There are a vast number of air compressors, air compressor models, and compressor shapes and sizes out in Do It Yourself Land. Yet all have similarities in plumbing that should make the advice on these pages useful to you.
For even more information (shameless plug alert) buy a copy of my e-Book The Home Compressor, which has chapters providing detailed information about how to connect up compressors, air tools, hoses etc., along with much more info on compressors.
The air tool might be a simple blow gun, air stapler, air brad nailer, or any of a myriad of air-driven, toys for big boys and girls that you can find gracing the shelves of your DIY stores these days. Regardless of the air tool type, the compressed air has to get to it.
The compressed air is prevented from flowing out of your compressor tank all the time by the discharge coupler threaded onto the end of the discharge pipe from the compressor.
In the photo the white circle is surrounding the discharge coupler installed on a small home compressor.
The air line to your discharge coupler from the tank may not look like that in the photo above, yet I assure you that the line from your tank in a typical DIY type air compressor will end in a coupler similar to that shown in the photo.
Here is more information about air couplers in general. I recommend you read this page for a complete understanding of couplers before proceeding further on how to connect an air line.
The something you insert into the coupler is the air connector, typically found either on the end of an air line, or in the supply port of an air tool.
In this photo the part of the connector that has been circled is the part that inserts into the coupler. Inserting the connector can be easy, or very challenging, depending on whether they are properly matched.
Sometimes it's necessary to slide the knurled ring on the coupler back as you insert the connector. Let go of the knurled ring before you let go of the connector so that the locking mechanism engages first. Otherwise, your connector can just blow back out of the coupler.
This is a male-NPT connector, and the part in the square box is the male thread to be threaded into the tool port, maybe a valve port, anywhere that you want the compressed air to flow when the connector is inserted into the coupler.