A transfer window is the period in which a club can register a new player with its league. Most countries run a long window in the summer and a shorter one in mid-season. Outside those dates a signing can be agreed but the player cannot be registered to play.
Registration is the thing that actually matters. The window does not stop clubs negotiating, agreeing terms or even signing contracts; it stops the paperwork that makes a player eligible. Deals are sometimes announced weeks before a window opens and completed on the first day it does.
Windows are set by each national association within limits agreed with FIFA, so the closing dates across Europe are close together but not identical. Leagues outside Europe run on their own calendars entirely, following their own season rather than the European one.
Free agents are the exception. A player without a contract has no registration to transfer, so most competitions allow them to be signed outside a window, which is why clubs hit by injuries in midwinter go looking at players who are already out of work.