Should I be running Bitcoin Core or Bitcoin-XT? I just read about Bitcoin Core and Bitcoin-XT, and I want to know what is the difference between Bitcoin Core and Bitcoin-XT?

And which is better Bitcoin Core or Bitcoin-XT?
The two pieces of software, Bitcoin Core and Bitcoin-XT, are very similar. Bitcoin-XT is actually just a fork of Bitcoin Core with some features added that Bitcoin Core developers didn't deem appropriate, such as a larger block size (BIP 101) and relaying of double spends.
Bitcoin Core is more up-to-date with all of the work that the community doing. Additionally, Bitcoin-XT now has a different consensus protocol regarding the maximum size of blocks. There has been much debate within the bitcoin community about if/how/when to increase the artificially imposed block size limit (of 1 MB), and one attempt to resolve this debate is being done through Bitcoin-XT.
The Bitcoin-XT README.md has these notes on how it is different than Bitcoin Core:


Support for larger blocks. XT has support for BIP 101 by Gavin Andresen, which schedules an increase from the one megabyte limit Bitcoin is now hitting.

Relaying of double spends. Bitcoin Core will simply drop unconfirmed transactions that double spend other unconfirmed transactions, forcing merchants who want to know about them to connect to thousands of nodes in the hope of spotting them. This is unreliable, wastes resources and isn't feasible on mobile devices. Bitcoin XT incorporates work by Tom Harding and Gavin Andresen that relays the first observed double spend of a transaction. Additionally, it will highlight it in red in the user interface. Other wallets also have the opportunity to use the new information to alert the user that there is a fraud attempt against them.

Support for querying the UTXO set given an outpoint. This is useful for apps that use partial transactions, such as the Lighthouse crowdfunding app. The feature allows a client to check that a partial SIGHASH_ANYONECANPAY transaction is correctly signed and by querying multiple nodes, build up some confidence that the output is not already spent.

DNS seed changes: bitseed.xf2.org is removed as it no longer works, and seeds from Addy Yeow and Mike Hearn are (re)added to increase seed diversity and redundancy.



So, if you support these changes and/or the possible future changes on the consensus-block-size-limit, run Bitcoin-XT. The releases are available here: https://github.com/bitcoinxt/bitcoinxt/releases