As a casual science reader, I've always found the implications of relativity (inconsistent clocks after near-light-speed travel and various space-time paradoxes) to be confusing and magical-sounding. Yet I know it's accepted as foundational to modern physics.
What are some of the experiments that have borne out Einstein's theory?
If you're asking for experimental tests of special relativity, arguably the first one is the Michelson-Morley experiment. Subsequent to that, there have been an enormous number of tests that verify it to incredible precision - a comprehensive account with references can be found here:
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html
But you've tagged your question with general relativity - for which there's a Wikipedia page that summarizes the main predictions (and confirmations) like the precession of Mercury's perihelion, deflection of light by the sun, etc.
You can find an extended list of experiments on the Special Theory of Relativity on the following link:
All the ones I know of are:
1) My personal favorite is the "atomic clocks flying around the world" test, or the Hafele–Keating experiment. Wikipedia summarizes it quite well:
The Hafele–Keating experiment was a test of the theory of relativity. In October 1971, Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took four cesium-beam atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another, and their differences were consistent with the predictions of special and general relativity.
2) Similarly, this also came up when the first atomic clock was put into orbit:
At the time of launch of the first NTS-2 satellite (June 1977), which contained the first Cesium clock to be placed in orbit, there were some who doubted that relativistic effects were real. A frequency synthesizer was built into the satellite clock system so that after launch, if in fact the rate of the clock in its final orbit was that predicted by GR, then the synthesizer could be turned on bringing the clock to the coordinate rate necessary for operation. The atomic clock was first operated for about 20 days to measure its clock rate before turning on the synthesizer. The frequency measured during that interval was +442.5 parts in 1012 faster than clocks on the ground; if left uncorrected this would have resulted in timing errors of about 38,000 nanoseconds per day. The difference between predicted and measured values of the frequency shift was only 3.97 parts in 1012, well within the accuracy capabilities of the orbiting clock. This then gave about a 1% validation of the combined motional and gravitational shifts for a clock at 4.2 earth radii.
3) Historically, the only obvious alternative to relativity (if I may oversimplify a bit) is the idea that light waves travel through some kind of medium, which was then called the aether. The Michelson-Morley experiment was an attempt to measure Earth's movement through the aether using a clever arrangement of lights and mirrors. Despite their best efforts, they never measured any such movement, which was a strong blow against the aether theory and set the stage for relativity. These lecture notes explain the full story in a thorough but approachable way.
4) People working with particular accelerators effectively re-prove this with every experiment. One of the simplest effects you have to take into account (i.e., one of the few I can claim to understand) is that firing one high-speed particle into a stationary particle takes a lot more energy than firing two slightly-less-high-speed particles towards each other.
5) The orbit of Mercury famously does not agree with Newtonian mechanics, unless one assumes there is another planet between it and the Sun. No one ever found such a planet, because it turned out general relativity was the correct explanation.
relativity [...] it's accepted as foundational to modern physics.
Correct.
What are some of the experiments that have borne out Einstein's theory?
There aren't any particular actual predictions contained in Einstein's theory to be "borne out by experiments" in the first place.
Relativity is comprehensible and accepted as foundational to (modern) physics from the outset (from "first principles"); for the very purpose of conducting any meaningful experiments and of obtaining any definite results at all (regardless of whether they might match some preconceived expectations, or not).
The notions and methods of relativity can be and are being communicated and understood by though-experimenental descriptions alone. It would be absurd to make the validity of definitions and procedures by which experimental results are to be gained to begin with in turn contingent on any particular result values thereby obtained.
Among the important experiments being set up, performed, and analyzed based on this foundation are
measurements whether any one given clock had been "good" (e.g. "ticking at regular rate") in any trial under consideration; or how to quantify if it did not,
measurements whether any two given "good" clocks had been equal to each other (e.g. "ticking" at equal "rates") in any trial under consideration; or how to quantify if they did not,
measurements whether pieces of given "experimental equipment" (e.g. "mirrors", such as the two "ends" of a given "optical resonator") had remained rigid to each other throughout any trial under consideration; or how to quantify if they had not,
measurements to quantify "mass", "stress", "energy", and/or "angular momentum" within a given region (e.g. the characterization of "Earth's interior" in terms of these quantites by the Gravity Probe B experiment; the determination of the distributions of those quantites within the Solar system, and beyond),
pretty much the entire identification of (known, or so far unknown) particles in experiments of particle physics.
None of their results are in any way "predicted by Einstein's theory"; therefore none of them could serve to "test or even to falsify Einstein's theory". Instead, the theory of relativity can and is being employed for obtaining those results in the first place.
I don't believe there are any plausible experiments which have proved the theory of relativity and I do not believe that time can pass slower on one body than it can on another. Imagine a train moving at 100km per hour relative to someone standing by the train tracks. A person on the train turns on a torch. The torch is on the train and would have the same speed of 100Km per hour relative to the person standing by the train tracks. However the beam of light is not formed within the torch. It only exists once it has left the torch so therefore it isn't really "on the train". The person on the train tracks would see that same beam of light exactly the same way as if the torch was being shone by a person standing directly beside them. Experiments have to be very well documented, describing all measuring equipment, measurements, margins of error, and they most importantly have to be repeated, preferably many times and by many different people. This has not been done for the theory of relativity. But it is now imbedded in our culture as being fact and no-one wants to come out and say that it always was a sham.
