Relativity is, like quantum mechanics, a fairly simple concept with incredibly complex implications and edge cases.
The easiest intuitive explanation of relative time I have found is the following:
- Assume our space time/universe is a 4D vector space. Three spatial dimensions, one time.
- Everything has a 4D vector which describes movement through this 4D space-time {x, y, z, t}
- The length/magnitude of this vector is fixed at the speed of light, c
- Normalize the dimensions of our 4D space as following: if we are not moving at all (x = y = z = 0) the vector points entirely in time our vector is {0, 0, 0, c} and, in our frame of reference, time moves as normal - "one second per second".
- Now, as we go faster and faster this vector no longer points purely in time. In the extreme case where the vector sum of the x, y, z (3D spatial component) is c, the time component t will be zero. This is why our photon does not experience time passing - it moves through space at c, so cannot move through time at all.
For the 3D velocities we normally experience we do not move in space fast enough to affect motion through time appreciably. Even the GPS corrections for relativity mostly arise due to Earth's gravity well distorting local space-time geometry, although their high velocity does contribute.
Objects with mass cannot be accelerated to c under our current understanding of relativity because kinetic energy diverges (approximately 1/sqrt(1-v^2/c^2)) - reaching speed of light would require infinite energy.