Router -> just routes packets between networks using the same protocol stack. Most of them are capable to do at all kinds of dynamic routing (like bgp, ospf, rip, isis and such). Generally you never see "simple" routers. Unless we talk about carrier grade equipment like CSR3, but even those have capabilities like NAT (carrier grade stuff though) which arguably are beyond the scope of a router.
L3 switch, like a router, but with more ports and less dynamic routing capabilities (generally they fully support intraAS protos like OSPF, but not interAS stuff like BGP). Generally they are also carrier grade stuff, and used for carrier's distribution network.
Gateway -> routes and translates packets between networks sometimes even if using different protocol stacks (IPV6 to IPV4, IPX to IPV4). Generally NAT capabilities are also considered part of a gateway, as they require some translations between routable and private network addresses. Most gateways lack things related to dynamic routing like BGP, OSPF or have very dumbed down stuff in them. The DHCP and DNS servers are generally also considered a "gateway feature". Arguably a gateway with only TCP/IP and NAT capabilities can be called a router.
WiFi HUB -> connects wireless clients in the same network (technically it is calld collision domain).
AP -> A WiFi hub that also bridges a wireless and a wired network to be seen as a single collision domain (translates frames between the two media in the same network, not routing packets between different networks). In some aspects are similar in role to media converters/gbic transcievers between optical and cooper media, or even multiplexers which allow you to transport a packet switched network over a circuit switched network.
Firewall : filters the packets that are passing trough it.
Switch : basically the same as ap, but for cooper. Has some improvements though over a basic hub though.
DHCP server : well, it assigns assigns IP addresses. Can be considered a part of the "gateway" .
Caching DNS server: it answers to DNS requests after going ito the upstream, and then caches the said DNS requests for fast retrieval. Can also be consideret part of the "gateway".
So your average SOHO WIFI router is arguably/technically not even a router, but the correct term would be "wireless capable TCP/IP residential gateway" or smth, as it is actually a gateway/switch/ap/firewall combo, with some servers (like caching DNS and DHCP servers) built into it.