well the twin paradox is about one twin leaving for a space trip (obviously VERY fast, 0,9+ of lightspeed) and the other twin staying at home.
although the earth isnt stationary, the space ship's time will differ greatly and when the astronaut comes back, he might look the same but his brother could be dead or an old man (depending on speed and length of the space trip).
relativity states that times goes slower near big masses or during fast movements (it might be the same since energy is used to accelerate a space ship and energy is mass so...), so its not a paradox, just something that feels like a paradox.
the barn paradox (or garage paradox) is very interesting. distances get also compressed/elongated during high speed travel. so, if you have a 50 ft long limousine, driving towards a regular 25 ft garage at 0,99 c, the limousine gets compressed (if seen from the garage) and it looks like it would fit into it. on the other hand, the driver would say that the garage became even smaller.
shit is very interesting when you think about a 20 ft car fitting into a 20 ft garage with the backdoor closed, the frontdoor open. the car drives into it, the frontdoor gets closed and the backdoor opens in this millisecond. its actually kinda interesting, because different observers would see the doors open/close at different times.
both "paradoxes" (they arent, as i said, real paradoxes, they just dont fit in our newton-based minds) can be "experienced" at much lower speeds, at least in theory. i guess that astronauts live a tad longer and that a car at 200 mph looks a thousandth of an inch shorter as when standing still.
i wrote all of this without looking anything up, so if i made a mistake, please correct. i know i confused some things. or look the stuff up on wiki, what i remember they are excellently described there.