Gauge theory is the collection of techniques used to describe forces in quantum field theory (electro-magnetism, weak force, and strong force). In field theory you have many vector spaces (1-d, 2-d, ... n-d) defined for every point in space time. So a specific field requires a distinct vector value for each space-time point. These fields encode certain particle types like electrons, quarks, etc.

One can ask, how can I compare field values at 2 distinct points a and b? They are in distinct vector spaces V_a vs V_b. Well, gauge theory says you must imagine the vector v_a being moved along a path between a and b, and see how the translated vector value compares to the vector v_b over b. This is called "parallel transport." A rule about how all vectors move along all paths during this translation is called a "connection." These connections encode distinct force-carrying particle types like photons and gluons. Sort of like how an object naturally rotates along a curvy surface. It's naively possible to describe this transport, this curvyness, but the natural notation is such that distinct descriptions are actually the same connection. Dealing with this ambiguity is key to properly computing probabilities using path integrals.

By the way, each gauge theory is required to have an underlying "gauge group," meaning a set of possible symmetries. In the standard model these groups are called U(1), SU(2) and SU(3). U(1) is 1-dimensional, so only 1 kind of electromagnetic force particle, the photon. SU(2) is 3-dimensional so 3 types of weak force particles (W-,W+, and Z) and SU(3) is 8 dimensional so 8 types of gluons.