The core idea is that each measurement you can make allows you to compute a curved surface on which the receiver must be located. Assemble multiple surfaces from different measurements, and look for the one point where all those surfaces intersect. Wherever that intersection is, your receiver must be at that point. There is an awful lot more to it than that, but that is the basic plan.
To locate a point in three dimensions, you need at least three constraints. If there are additional unknowns for which you would also like to solve, you need more constraints. GPS uses a minimum of four because you have to solve for not only your location, but also the time at which the measurements were made, because including a clock good enough to give decent answers by itself in the receiver hardware would make the things cost tens or hundreds of thousands of dollars and weigh hundreds of kilograms (chip-scale atomic clocks are starting to change this, but GPS was designed with 1970s hardware in mind).
Sometimes multiple measurements tell you to compute the same surface; this redundancy means you don't have enough surfaces until you get more some other way (waiting and trying again, measuring a signal from a different source, measuring a different aspect of a signal, etc.), so you can intersect them to find out where you are. Of course, in the real world, there is measurement error, so you will always be trying to solve slightly the wrong equations. This means there probably won't be one point in which all the surfaces intersect, because you've got the wrong surfaces. The standard solution is to overdetermine the system: make not the smallest number of measurements you can get away with, but make as many as you can in the time available, and use a mathematical optimization algorithm to find the one location most consistent with the whole set of observations. You must know the locations of some references, so that the measurements you make of signals sent from them can be converted to position; there is always some error in that knowledge. If your reference sources are moving (and if they are in orbit, they are moving very fast), you need to know a lot more, have data that's kept extremely current, and still have to live with a sometimes considerable amount of residual error due to bookkeeping all of that.
If you are measuring range from a known point, then your receiver must be located on the sphere with that radius centered on that known point (GPS is usually described this way, but since they have to solve for the clock bias, it's really closer to the LORAN case under the hood). If you are measuring time differences between two signals from two known points (as in LORAN), then your receiver must be located on the hyperboloid with foci those two points and asymptotic cone angle corresponding to the measured time difference. If you are measuring the Doppler shift applied to a signal from a fixed location by your satellite's orbit (the DORIS system), the surfaces are more complicated than conic sections, and depend on your velocity as well as your position, so you need to solve for at least six dimensions. If you are measuring the frequency difference of arrival between two different signals, the surface is given by an eighth-degree polynomial, having level sets that sometimes look like a banana, or a peanut, or a donut, or two pointy hats on a table, or a wide variety of other things.
Additional complications come from things like atmospheric propagation. For example, since the atmosphere refracts radio waves, they don't travel in straight lines, so the surface of constant travel time from emitter to receiver is not actually spherical. In GPS work, if you have a dual-frequency receiver, you can use the difference between the propagation effects on the two frequencies to infer total electron content along the signal travel path, and use that to figure out what the undistorted signal might have been if you were able to receive it without interference. This just scratches the surface of a very broad area of knowledge.