Relativity

It's difficult to provide a good summary of relativity. Volumes have been written and, in the end, the math only gets harder. It can be explained as 'everything is relative', but that does not properly convey the magnitude of what is going on. If you're -really- interested, see Relativity on the World Wide Web, which is a cool source of links period.

Beginning with simple concepts like Newton's laws of motion, imagine an object in a vacuum. Assuming no acceleration is occurring, it perceives itself to be at rest no matter how fast you see it moving, just as you perceive yourself to be at rest without experiencing acceleration. And, no matter how fast you perceive something to be moving, its speed will not exceed that of light. This same is true for every single object in the universe. As a consequence of this, things moving at the speed of light (like say, light) are perceived to move -at- the speed of light by every potential observer.

In order for this to make any sense, it must also be true that the passage of time itself is -also- relative. Skipping most of the math, imagine two people are moving away from or towards each other at about 86% of the speed of light, and they're each wearing a watch. Both of them, of course, see their watches progress at a normal rate. Each one, however, sees their partner's watch ticking off the seconds at half the normal rate.

That is, when person A counts of 60 seconds, she sees person B count off 30 seconds. While person B has counts off 30 seconds, he sees person A count off 15...

Progression through time, too, is relative. There is no such thing as a universal 'now' in which all observers, or even two observers, can really agree upon. This is where the idea that superluminal travel equals time travel comes from. 'Causality, FTL, Relativity, pick two.' is a popular axiom.

For a science fiction setting that does not permit time travel and needs faster-than-light speeds, this is a problem. There are several potential solutions. The first thing to keep in mind is that relativity's prohibition of superluminal motion comes specifically from Special Relativity - a localized case that ignores the curvature of spacetime.

1. Keep all motion within its own light-cone. Black holes and the expansion of space permit forms of relatively superluminal motion, but these events are hidden from an observer behind horizons, as they occur due to the motion of spacetime. The Alcubierre metric makes use of this trick, at the cost of violating several energy conditions.
2. Change the speed of light. While another popular option, this exact effects of this are not easy to predict. This would certainly wreak havoc with the fine structure constant, which governs the charge of the electron.
3. Wrap relativity in a higher theory that constructs a preferred frame of reference in such situations. While this can allow for many possibilities, it is entirely a fantasy mechanism.

A trap many schemes tend to fall into is measuring velocities in some number 'times the speed of light'. This is no improvement - saying it will get you from Sol to Alpha Centauri in a day is meaningful, but this is only ~1,570 times c in our own frame of reference. In fact, for something moving at 99.99998% the speed of light, going from Earth to Alpha Centauri in a day is slower than light.

Solar Storms

Solar Storms blends all three points above, though mostly the first and third. Obviously, it is assumed that Minkowski spacetime is inappropriate for describing these situations.

• Stargates function via the first method - they warp space in such a manner that, with no change in vector, it is not possible for a ship transversing one to outrun a photon. The mouths of a stargate have 'spacial bleed' which must be skirted in order to actually move from one star system to the next at 'faster than light'.
• Tachyons work through a preferred frame of reference. The active speed appears to be instantaneous, or nearly so. Non-comoving observers will not always agree on their speeds when they can be determined.
• The technology does exist to locally change the speed of light. Within the Universe, this is not found to be particularly useful for transportation or communication purposes.
• Regalic vessels are capable of levels of energy output that make many other forms of transport possible and make use of all three.

In these scales, a different form of relativity still remains, though it sees far less use.