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As a general overview, I should mention that General Relativity does have concept(s) that relate to "frames of reference", but there are some important differences. It's hard to be precise in lay language, but the biggest change is that what passes for "frames of reference" in GR are for the most part purely local. So to give an example, if one is in the MIR space-station in Earth orbit, one can construct a local frame of reference that "moves with" the space-station and covers the section of space-time within the space-station without significant problems. Trying to cover too much area (an entire orbit) with a single frame of reference tends to cause issues, though.
So if one is basing all of one's physical understanding on the existence of a frame of reference, one may feel at a bi of a lost, since they don't quite work the same way in GR. What does exist is the concept of a coordinate system. In general the properties of a coordinate system are pretty basic, the only requirement is a unique mapping from every physical event (a point in space at a specific time) in the region covered by the coordinate system (which often covers all of space-time, but this is not actually required) to a set of numbers, called "coordinates", that tell one when and where the event occured. This is a more basic, more primitive construct that represents the structure of space-time than a "frame of reference".
Getting things like distances out of these generalized coordinates is more involved, however, the mathematical tool that gets distances and/or time intervals out of coordinates is called a metric.
So if one is basing all of one's physical understanding on the existence of a frame of reference, one may feel at a bi of a lost, since they don't quite work the same way in GR. What does exist is the concept of a coordinate system. In general the properties of a coordinate system are pretty basic, the only requirement is a unique mapping from every physical event (a point in space at a specific time) in the region covered by the coordinate system (which often covers all of space-time, but this is not actually required) to a set of numbers, called "coordinates", that tell one when and where the event occured. This is a more basic, more primitive construct that represents the structure of space-time than a "frame of reference".
Getting things like distances out of these generalized coordinates is more involved, however, the mathematical tool that gets distances and/or time intervals out of coordinates is called a metric.