You're describing a wireless bridge between the RPi's Ethernet and wireless interfaces. Simply using normal Ethernet Layer 2 bridging (e.g. bridge-utils) won't work due to wifi design limitations, however you can achive the same effect via Proxy ARP. Proxy ARP is a routed (Layer 3) solution, but works for all IP traffic: see https://wiki.debian.org/BridgeNetworkConnectionsProxyArp for details, but it is as simple as:
$ sudo apt-get install parprouted dhcp-helper avahi-daemon
Edit /etc/network/interfaces to configure the interfaces:
auto lo
iface lo inet loopback
auto eth0
allow-hotplug eth0
iface eth0 inet manual
auto wlan0
allow-hotplug wlan0
iface wlan0 inet dhcp
wpa-conf /etc/wpa_supplicant/wpa_supplicant.conf
post-up /usr/sbin/parprouted eth0 wlan0
post-down /usr/bin/killall /usr/sbin/parprouted
# clone the dhcp-allocated IP to eth0 so dhcp-relay will relay for the correct subnet
post-up /sbin/ip addr add $(/sbin/ip addr show wlan0 | perl -wne 'm|^\s+inet (.*)/| && print $1')/32 dev eth0
post-down /sbin/ifdown eth0
Edit /etc/sysctl.d/local.conf to enable IP forwarding:
net.ipv4.ip_forward=1
Enable DHCP relay: /etc/default/dhcp-helper
# relay dhcp requests as broadcast to wlan0
DHCPHELPER_OPTS="-b wlan0"
Edit /etc/avahi/avahi-daemon.conf to enable mDNS relaying:
[reflector]
enable-reflector=yes
Reboot, and hosts connected to the bridge's ethernet should acquire a DHCP address and have full IP connectivity!
You can connect another router to the Pi's Ethernet, in which case you'd very likely want to use that router's WAN port connected to the Pi (dont use the LAN ports - you're likely to end up with duelling DHCP servers at the least). If all you're doing is connecting multiple hosts via the Pi, just use a switch.