Oblivious transfer: a cryptographic two-party protocol, where either(a) the receiver gets sender's bit with probability $1/2$, without the sender getting to know whether the receiver got the bit; or (b) the receiver gets exactly one of sender's $N$ $\ell$-bit inputs, without the sender getting to know which one the receiver got; this is usually called an $1$-out-of-$N$ oblivious transfer protocol. In the $1$-out-of-$N$ PIR, receiver receives one of the inputs (and possibly more), without sender getting to know what receiver asked for.
One distinguishes between information-theoretically sender-private PIR and computationally sender-private PIR (CPIR). The first kind takes at least linear communication (in $N$) in the case of one server; there are sublinear protocols in the case of multiple servers. The second kind can be done much more communication-efficiently. In particular, [Lipmaa04] proposed a protocol with communication $O(\log^2 N+\ell\cdot\log N)$, based on the Decisional Composite Residuosity Assumption, and [GentryRamzan05] proposed a protocol with communication $O(\log N+\ell)$, based on a variant of the $\Phi$ Hiding assumption. In addition, in Lipmaa's protocol, receiver's work is polylogarithmic in $N$ while in the Gentry-Ramzan protocol, receiver's work is $O(\sqrt{N})$. For very large blocks, Lipmaa's protocol eventually achieves throughput $1$ (i.e., the CPIR communicates $\approx \ell$ bits) while the Gentry-Ramzan protocol achieves throughput $1/4$. Different flavors of PIR/OT give in general rise to very different protocols.
For a long time it was thought that one can construct an oblivious transfer protocol based on any trapdoor permutation. This is however not known to be true. [Haitner04] showed how to construct OT based on ``dense trapdoor permutations''. See [KushilevitzOstrovsky2000] for a slightly sublinear-communication OT (more precisely, PIR) based on one-way trapdoor permutations.
OT is complete for MPC by a classical result of Kilian.
A lot of work has also been done on quantum OT.
See [Rabin81] for the roots of oblivious transfer, Gasarch's link farm for a somewhat out-dated overview of (C)PIR.