What kind of contacts will we be seeing eventually on LN? Will it be anything like what Ethereum has?
(This is according to my understanding, so any comments are very welcome.)
Lightning Network Summary
Currently, the lightning network operates by first creating a transaction on the blockchain, this is called a channel. See LND Overview.
When someone does a transaction on the lightning network, they create a transaction that they sign, which could be signed by the recipient and "closed out" at any time, which broadcasts the final state of the channel to the blockchain.
The fact that either party can close the channel at any time by broadcasting it to the blockchain means there isn't any loss of trust. With this level of trust, this limits the transactions on the lightning network to the current Bitcoin Script System which is intentionally not turing-complete.
Theoretically, another layer could be built on Lightning network (or the Bitcoin network for that matter) that established it's own form of scripting (e.g. some turing-complete scripting like Ethereum), and then generated Bitcoin-compatible transaction scripts. But, then the user would have to trust the new scripting language to generate a valid transaction that is in their favor. However, that introduces a very large attack vector and possibilities for developer errors like we've seen on Ethereum smart contracts. This does not depend on the Lightning network, though some of the concepts of LN might be useful.