A router is most peoples "gateway" to the internet. It handles everything you need to get onto the internet. PPPo(A/E), DHCP, NAT as well as wireless connection and ethernet ports.
Bridge is a "grey" term and can be used in many contexts.
A wireless bridge is effectively 2 wireless devices connecting to each other forming an invisible cable between them.
A bridge on an interface is binding the 2, you may for example bridge 2 ethernet ports on your motherboard so one is connected to the router (providing internet) and the other to your xbox so it can get online.
"Bridge" is also a term used in some router configs in place of simply using "wireless access point" Usually when set as "bridge" a router becomes purely a wireless access provider and does not do any of its normal stuff (NAT, DHCP, sessions) and takes its connections from a router which is already providing all of this.
So to answer your last question, there is usually no difference between a router set as a bridge and a wireless access point, however it depends on the access point you are referring as they are a whole new catacomb of confusion.
If I am right from my "skim" over your other thread, use your current router "a" as a router and your second router "b" as a bridge and you will increase wireless connectivity through your house but there will need to be a cable between them.
I don't *think* or know of any router that can currently be a receiver over a wireless bridge AND broadcast it's own wifi network.
HTH