Shop welding is, or can be, a relatively controlled environment, so almost any process can be used in the shop, depending on the application. The most common processes used for mild steel shop welding (like most building structural fabrication) are Shielded Metal Arc Welding (SMAW), commonly called "stick" welding and Gas Metal Arc Welding (GMAW), commonly called "MIG" welding. One variation of the GMAW process is Flux Core Arc Welding (FCAW), using a similar wire feed process but having protective flux within the core of the wire so that inert gas shielding, as used with GMAW and GTAW, is less critical and can be deleted in many cases, particularly for field use.
For large, automated welding applications (such as fabricating plate girders or tanks), Submerged Arc Welding (SAW) is commonly used in the shop.
Other shop processes are Electric Resistance Welding (ERW) for welding seamed pipe and similar applications, Gas Tungsten Arc Welding (GTAW) commonly called "TIG" welding, and a variety of other specialized processes. GTAW is used for finer welding of thin sheets, welding non-ferrous metals like aluminum, and for making root passes of critical welds.
Field welding is most commonly done using the SMAW process. It requires no wind breaks, is done with relatively simple equipment and the welding variables can be controlled more easily than other processes. FCAW can be used in the field. GMAW can also be used in the field provided wind and drafts can be controlled so that shielding gas is not blown away. GTAW has similar issues to GMAW when used in the field.
A couple of good references....AWS Structural Welding Code-Steel (AWS D1.1) and other AWS structural codes, and a primer on welding that was published by the Lincoln Welding Foundation titled "Metals and How to Weld Them".
That's a very complete and correct answer, given by Ron. not much too add there.
I'd just say try to avoid GMAW and GTAW in the field, or use the necessary precautions as the shielding is very important.
Also, be aware there's still a few dozen other processes available, some a bit more exotic than others.
