Ok, they’re fundamentally a pretty simple idea. A DHT gives you a dictionary-like interface, but the nodes are distributed across the network. The trick with DHTs is that the node that gets to store a particular key is found by hashing that key, so in effect your hash-table buckets are now independent nodes in a network.
This gives a lot of fault-tolerance and reliability, and possibly some performance benefit, but it also throws up a lot of headaches. For example, what happens when a node leaves the network, by failing or otherwise? And how do you redistribute keys when a node joins so that the load is roughly balanced. Come to think of it, how do you evenly distribute keys anyhow? And when a node joins, how do you avoid rehashing everything? (Remember you’d have to do this in a normal hash table if you increase the number of buckets).
One example DHT that tackles some of these problems is a logical ring of n nodes, each taking responsibility for 1/n of the keyspace. Once you add a node to the network, it finds a place on the ring to sit between two other nodes, and takes responsibility for some of the keys in its sibling nodes. The beauty of this approach is that none of the other nodes in the ring are affected; only the two sibling nodes have to redistribute keys.
For example, say in a three node ring the first node has keys 0-10, the second 11-20 and the third 21-30. If a fourth node comes along and inserts itself between nodes 3 and 0 (remember, they’re in a ring), it can take responsibility for say half of 3’s keyspace, so now it deals with 26-30 and node 3 deals with 21-25.
There are many other overlay structures such as this that use content-based routing to find the right node on which to store a key. Locating a key in a ring requires searching round the ring one node at a time (unless you keep a local look-up table, problematic in a DHT of thousands of nodes), which is O(n)-hop routing. Other structures – including augmented rings – guarantee O(log n)-hop routing, and some claim to O(1)-hop routing at the cost of more maintenance.
Read the wikipedia page, and if you really want to know in a bit of depth, check out this coursepage at Harvard which has a pretty comprehensive reading list.