Finding it by hand
Take the network address and set every host bit to 1. For 192.168.1.0/24 the host part is the last octet, so the broadcast address is 192.168.1.255. For a /26 that starts at 10.0.0.64 it's 10.0.0.127, and the hosts in between run 10.0.0.65 to 10.0.0.126.
The first address of a block is the network address and the last is the broadcast address. Neither goes to a device, which is why a /24's 256 addresses give 254 hosts.
Limited and directed broadcasts
A packet sent to 255.255.255.255 is a limited broadcast. It reaches everything on the local link, and routers never forward it. A packet sent to a subnet's own broadcast address, like 192.168.1.255, is a directed broadcast. Routers could forward those into the target subnet, but RFC 2644 says that option must be off by default, so a directed broadcast from another network almost never arrives.
Broadcasts still do real work inside a network. ARP uses Ethernet broadcasts to ask which device owns an IP address. DHCP clients broadcast to find a server before they have an address of their own.
Where there's no broadcast address
RFC 3021 lets a /31 link use both of its addresses as hosts, with no network or broadcast address. That suits a point to point link between two routers. The calculator counts 2 usable addresses at /31 and 1 at /32, a single host. A /30 spends 4 addresses on the same two hosts.
Cloud networks don't carry broadcast at all, yet AWS still holds back the last address of every subnet, and Azure blocks broadcast traffic. So a /24 subnet in AWS has 251 usable addresses: 256 total minus five.
IPv6 dropped broadcast completely. RFC 4291 states that IPv6 has no broadcast addresses, and multicast does the job instead. The address ff02::1 reaches every IPv6 node on the local link.