Help in understanding exactly how lattices used as one way functions for hashing
You have several confusions regarding cryptography. First, the nature of hash functions. The non-cryptographic use of hash functions is to hash long messages into shorter ones in a "uniform" fashion. So we expect there to be many collisions, by design. Cryptographic hash functions have the additional properties that collisions are hard to find (there are several, inequivalent ways of stating this). Therefore, while it is possible to find collisions even for cryptographic hash functions (simply because the range is smaller than the domain), this should be difficult. Such hash functions can be used in various ways in cryptography, for example in digital signatures: you sign on the (small) hash rather than the (large) message.

Second, encryption is a different primitive from hash functions. Encryption itself comes in two main kinds, symmetric and public key, which are rather different, and there are many other cryptographic constructions out there. There are reductions between some of the cryptographic "primitives" (hash functions, one-way functions and the like), for example you can use them to construct HMACs (message authentication codes) and PRNGs (pseudorandom number generators), but perhaps not encryption schemes.

Regarding the matter at hand: the idea of Ajtai's hash function is to use a random lattice $A$, the hash function being $x \mapsto Ax \pmod{q}$, for some prime $q$. Here $x$ is a binary vector in $\{0,1\}^m$, embedded into $\mathbb{Z}_q^m$. The parameters here are important &mdash; the size of $A$ and $q$ &mdash; but I'll ignore them here. In order to show that this is cryptographically secure, under one definition we need to show that it is difficult to find a collision, that is two messages $x,y$ with the same hash. Such messages would satisfy $A(x-y) \equiv 0 \pmod{q}$, and so $x-y$ would be in the dual lattice $A^{\perp}$. Now $x-y \in \{-1,0,1\}^m$, so (for the correct choice of parameters) it is "short", and finding it is conjectured to be difficult. This shows the (conditional) security of the scheme.

Lattice-based encryption can also be used for public-key encryption (Regev's scheme) and for homomorphic encryption. These schemes rely on a different problem, LWE (learning with errors), which is similar to CVP (closest vector problem). The details are more complicated &amp;mdash you can have a look at Regev's scheme in Section 5.4 of his survey with Micciancio, though it could be hard to glean from there why exactly the scheme works. Perhaps this is explained later in your course, and if you need more explanations, you can ask another question here.