So, is anyone able to explain to me the twins paradox?
I personally don't know why they call it a paradox. Maybe it's because they assume that it's just relative speed that should cause each twin to time-dilate as observed by the other, i.e. if two twins, are made to pass each other at a significant (but constant) fraction of the speed of light (and are considered 'synchronised' immediately before they pass) then they would see each other age less than themselves. And the same would apply whether one twin were moving or both. To the view of an external observer, who may or may not be moving themselves, you can never really say that you are stationary, just work on the assumption that you are if you are not accelerating in any way.
Assuming the twins are given identical 'acceleration patterns', in their respective round-trips, then when they meet up again, they will remain synchronised. For each twin, the altered perspective of time passing for the other twin when the distance is widening retreating will (to cut a long story far too short) reverse itself when they distance narrows again. Kind of like how the doppler effect on a siren on an emergency vehicle would mean that you'd end up hearing exactly the right amount of warbles overall, as you rush away from it and then it catches back up to you (assuming travelling below the speed of sound), although you'd get a lower frequency than normal at first, then a higher than normal one later on. (This would also apply to a wrap-around universe, as the twins see each other converging from the opposite direction that they departed, with the caveat that there's probably still a 'delayed' image visible in the direction of departure, which eventually converges with the delayed 'echo' of themselves, should you be able to see that far.)
But with reletivistic effects, it's also acceleration that's the crux. Put (a little too) simply, acceleration with respect to a frame of reference is the additional key. Usually looked at as linear acceleration, but change of velocity by turning to a different heading or acceleration due to gravity while otherwise standing 'stationary' on the surface of, e.g., a planet also count.
With the twins, the twin on the planet experiences just the (trivial, almost inconsequential) acceleration of gravity. The twin who is sent to Alpha Centuri and back in (say) a 10 year round trip (1 and 1/4 years longer than light would take, IIRC) might experience accelerative dilation to a factor of roughly (if I've not messed up in the relevent mental arithmatic) 1/4. i.e. 2.5 years passes for the acceleration-affected twin, the one who had to somehow accelerate fast enough to get to A-C and back (including deceleration half way through each journey, which is still an acceleration, just in the opposite direction).
If it's a 100-year round trip, for the 'just round the corner' trip to Alpha Centuri, then (human longevity aside) we're talking about near 99% time synchronisation, because the twin hasn't needed to accelerate much. (I could work out what travel time a 1G acceleration/deceleration would give, but I'm already waffling too much and something's niggling me that I've got the equations a little skewed anyway. Still, it's representive if not strictly accurate.)
I think probably what you want to look at, Twins Paradox-wise, is the experiments in which the atomic clock was flown round the world and compared with its 'twin' left stationary on the ground. IIRC, the ground-based clock experienced the small acceleration-due-to-gravity dilation, as normal, while the flying one marginally less of that (due to average altitude being greater) but had a dilation effect of its motion. Each effect was very small, but even when combined in opposition it was detectable by the accurate timepieces.
And of course moving (or rapidly 'jiggling', I suppose) one end of a wormhole while letting the other stay at (relative) rest is a classic method of giving you a time machine. Wormhole physics allowing, of course.