XYZ is a convention whereby the first 3 bids of a partnership's auction are on the 1-level. After those 3 bids (1X-1Y-1Z), the typical treatment is to use:
2 by responder as an artificial relay. Opener must bid 2.
2 by responder as an artificial bid, saying that the partnership is going to at least game.
Other bids by responder are natural and NOT Forcing; 2-level suit bids are typically weak, 2NT and 3-level bids are invitational.
Responder's 2 relay is used to either place the contract in 2, or to be followed by an invitational bid.
The first 3 bids can be any 3 bids on the 1-level -- including 1NT.
The first 3 partnership bids are on the 1-level, so the convention applies.
Now, the responder can bid 2 to force opener to bid 2. After this, responder can pass to play 2, or make an invitational bid (such as 2, which would promise at least 5 spades). Notice that this is a variation on the better known "new-minor checkback", or "new-minor forcing."
Over 1NT, responder can also bid 2 (artificial) to set up a Game Force. After this, opener bids naturally--he would raise the major with 3-card support.
Responder can also directly rebid 2-of-either-major (weak), 2NT invitational, or bid on the 3-level (invitational).
Notice that there are several ways to invite (via 2 first, or sometimes directly). Serious partnerships can invent their own meanings for the difference. For example, define a different meaning for 1-1-1NT-2NT and 1-1-1NT-2-2-2NT. Perhaps going through the relay of 2-2 is a 5-3-3-2 invitation with 5, whereas the direct 2NT denies 5.
Again, the responder can bid 2 to force opener to bid 2. After this, responder can pass to play there, or make an invitational bid (such as 2, which would promise at least 5 hearts). Responder can also bid 2 to set up a Game Force.
Other actions as above.
Nothing changes. Responder can relay with 2 or start a Game Force with 2. In both cases, of course, the 2-of-a-minor bid is completely artificial. Note: Since almost everyone reading this article is likely to be using "Fourth Suit GF," be aware that you are still using it, but the "4th suit" will sometimes not really be the 4th suit (because after 1-1-1, the artificial GF action is now 2, not 2).
If using XYZ, it does not matter what the first 3 bids were, as long as they are all on the 1-level. The auction could even begin 1-1-1. (In that case, 1 would be natural and 2 would be the artificial GF).
Of course, the major downside (other than forgetting) is that the partnership can't play in a 2 contract. The responder can signoff in three of either minor by jumping there after the 1NT rebid.
As with any convention, the partnership must decide if this is on in competition (I recommend NO) or by a Passed Hand (I recommend NO).
This treatment has been steadily gaining in popularity since the turn of the millennium.