I understand that free neutrons pervade the universe and that they are relatively unstable. What do they decay into?
A neutron is not a fundamental particle, rather it is made of three particles known as quarks bound together. There are six types (or 'flavours') of quark and a neutron consists of one so-called 'up' quark and two 'down' quarks.
When not bound in a nucleus, a neutron is unstable and one of the down quarks undergoes a beta decay (like in some radioactive nuclei) in which it becomes an up quark - this remaining arrangement of 2 up quarks and one down quark is a proton. Therefore, a neutron decays into a proton and this process also emits an electron and an anti-neutrino (a very light, uncharged fundamental particle). The half-life of a free neutron (if you were to have a collection of them, the time it would take for half of them to decay) is around 10 minutes.