The weak interaction, which converts protons into neutrons in the sun, allowing heavier elements like iron to be built from protons, is left-handed.
A proton is converted to a neutron via the weak force carried by the W boson.
The direction of time is up. One can see a u quark changing to a d quark, by emitting a W boson, which then decays into a positron and a neutrino.
The handedness refers to which direction it is spinning, compared to its motion. Certain interactions happen only to particles spinning in one direction. This strange result is like you are dancing in front of a mirror, and your reflection is just standing there, doing nothing.
An unsolved mystery today is that neutrinos appear only to exist as left-handed particles. Right-handed neutrinos have never been observed.
But just because we've never seen right-handed interactions, doesn't mean we don't look. My PhD thesis was to search for right-handed interactions in top quark decay. The idea is that perhaps the top quark, the heaviest known particle, is weird, and doesn't behave like neutrinos. Unfortunately, my thesis disproved the idea that the top quark has right-handed decays. More information here.