A form A contact is single throw, normally open, closes when relay is operated.
A Form B contact is single throw, normally closed, opens when relay is operated.
A form C contact is double throw - the moving contact is connected to one "output" when the relay is released, and to another when the relay is operated.
simulate this circuit – Schematic created using CircuitLab
(Our handy schematic editor doesn't have a Form B switch)
A single pole relay will have only one set of switch contacts. Double Pole will have two sets, so can switch two circuits. Relays are available with four or more poles.
The Form A, B, and C terminology is also used with switches. Form C contacts may be described as "shorting" or "non-shorting". In a "sorting" switch, all three terminals will be connected together briefly as the switch moves between positions.
Relay and switch contact arrangements are often given as "SPST-NO" (Single Pole, Single Throw, Normally Open) for a single Form A contact, or "DPDT" (Double pole, Double Throw) for two form C contacts.
A Double Throw (form C) switch or relay can be substituted for either Form A or Form B - just ignore the unwanted contact.
I'm not familiar with X, Y, or Z contact arrangements.