Quick answer
An AAAA record maps a hostname to a 128-bit IPv6 address, and is the IPv6 counterpart of the A record that carries a 32-bit IPv4 address (RFC 3596 §2.1). It is pronounced "quad-A". A single record looks like example.com. 3600 IN AAAA 2001:db8::1. Most public hostnames that support IPv6 publish both an A record and an AAAA record so clients can reach them over either protocol.
What an AAAA record contains
RFC 3596 defines the AAAA resource-record type, IANA-assigned numeric type 28, to store one IPv6 address as 16 octets in the record's RDATA (RFC 3596 §2.1–2.2). The "four-A" name reflects its size relative to the A record: an A record holds a 32-bit address, and an AAAA record holds an address four times as long.
The value is written using the standard IPv6 presentation format (RFC 3596 §2.4), where a single :: stands in for one run of consecutive all-zero groups. Example zone lines:
example.com. 3600 IN AAAA 2001:db8::1
mail.example.com. 3600 IN AAAA 2001:db8:0:1::25
A name can carry more than one AAAA record if the host answers on several IPv6 addresses. The resolver returns the full set, and the client picks one to connect to.
AAAA and A records: dual-stack
Publishing an A record and an AAAA record for the same name is called dual-stack. The name resolves to an IPv4 address for a client that queries for type A, and to an IPv6 address for a client that queries for type AAAA. Nothing links the two records inside DNS; they are independent entries that share a name.
A client that can use both protocols does not choose at random. Under the "Happy Eyeballs" algorithm in RFC 8305, the client requests the A and AAAA records in parallel, starts connection attempts on the addresses it receives, and keeps whichever connects first, with a preference for IPv6. This is why a wrong or unreachable AAAA record can make a site feel slow for some visitors while it stays fine for others: their client tried the IPv6 address first and had to wait for that attempt to time out before falling back to IPv4.
IPv6 reverse DNS and the PTR record
A forward lookup answers "what address does this name have". A reverse lookup answers the opposite question, "what name owns this address". IPv6 reverse DNS lives under the ip6.arpa tree (RFC 3596 §2.5). To build the reverse name, the address is expanded to all 32 hexadecimal digits, those digits are reversed and dotted, and ip6.arpa is appended; a PTR record at that name then returns the hostname. The reverse name for 2001:db8::1 is:
1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.8.b.d.0.1.0.0.2.ip6.arpa
The PTR record page covers how that mapping is delegated and populated.
AAAA records and email deliverability
A server that sends mail from an IPv6 address effectively must have working reverse DNS for that address. Google's and Yahoo's bulk-sender guidelines require every sending IP to have a valid PTR record whose hostname also resolves forward to the same IP, and that requirement covers IPv6 senders. Microsoft is stricter still: Outlook.com rejects inbound IPv6 mail when the sending IP has no reverse DNS entry at all.
An IPv6 mail server therefore needs the forward AAAA record for its own hostname together with a matching PTR record for the IPv6 address it connects from. You can confirm the reverse side with the reverse DNS lookup tool. If you cannot guarantee a correct IPv6 PTR for a given sending host, a common choice is to leave that host off IPv6 for outbound mail and send over IPv4, where the reverse DNS is already under your control.
Do you need an AAAA record?
Add an AAAA record when the host has a reachable IPv6 address and you want IPv6 clients to connect over IPv6. Website and API hostnames served through an IPv6-capable CDN or load balancer usually get one provisioned for you.
Leave the AAAA record off when the address behind it is not actually reachable over IPv6. A stale AAAA record that points at an IPv6 address which no longer answers is worse than having none, because Happy Eyeballs clients try it first and stall on the dead attempt. For a mail-sending host, publish the AAAA record and start sending over IPv6 only once the matching PTR is in place.