Does gratuitous ARP work like a normal ARP request?

Why is gratuitous ARP used for HSRP?

Gratuitous ARP is a sort of "advance notification", it updates the ARP cache of other systems before they ask for it (no ARP request) or to update outdated information.
When talking about gratuitous ARP, the packets are actually special ARP request packets, not ARP reply packets as one would perhaps expect. Some reasons for this are explained in RFC 5227.
The gratuitous ARP packet has the following characteristics:

Both source and destination IP in the packet are the IP of the host issuing the gratuitous ARP
The destination MAC address is the broadcast MAC address (ff:ff:ff:ff:ff:ff)

This means the packet will be flooded to all ports on a switch


No reply is expected

Gratuitous ARP is used for some reasons:

Update ARP tables after a MAC address for an IP changes (failover, new NIC, etc.)
Update MAC address tables on L2 devices (switches) that a MAC address is now on a different port
Send gratuitous ARP when interface goes up to notify other hosts about new MAC/IP bindings in advance so that they don't have to use ARP requests to find out
When a reply to a gratuitous ARP request is received you know that you have an IP address conflict in your network

As for the second part of your question, HSRP, VRRP etc. use gratuitous ARP to update the MAC address tables on L2 devices (switches). Also there is the option to use the burned-in MAC address for HSRP instead of the "virtual"one. In that case the gratuitous ARP would also update the ARP tables on L3 devices/hosts.
