What is the most efficient lattice problem solving algorithm? I've recently become very interested in post-quantum cryptography, specifically lattice-based cryptography. As of this posting there exists no quantum algorithm that can perform better at solving lattice problems than a conventional computer.

There are, however, algorithms that exist that theoretically can solve lattice problems, albeit slowly.

I'd like to study one or more of these algorithms myself to help me better understand lattice cryptography and lattices in general.

So, what is the most efficient lattice problem solving algorithm and why is it so efficient?
What you want is probably a lattice reduction algorithm.

Lattice cryptography usually relies on the Closest Vector Problem (CVP) or the Shortest Vector Problem (SVP). Solving these involves the Lenstra–Lenstra–Lovász lattice basis reduction algorithm, or LLL. This algorithm is the quintessential lattice reduction algorithm, and is probably what you are looking for. This algorithm is used internally by other algorithms, such as Babai's nearest plane algorithm for CVP.

Note that the LLL algorithm is not able to reduce lattices in higher dimensions, which is the case with practical lattice cryptography. Instead, different algorithms are required, such as BKZ 2.0 for NTRU. However for the purpose of learning about lattice reductions, understanding LLL should be sufficient.