NAT EXPLANATION
Network Address Translation (NAT) is the simplest way of accessing an external network from a virtual machine. Usually, it does not require any configuration on the host network and guest system. For this reason, it is the default networking mode in VirtualBox.
A virtual machine with NAT enabled acts much like a real computer that connects to the Internet through a router. The “router”, in this case, is the VirtualBox networking engine, which maps traffic from and to the virtual machine transparently. In VirtualBox this router is placed between each virtual machine and the host. This separation maximizes security since by default virtual machines cannot talk to each other.
The disadvantage of NAT mode is that, much like a private network behind a router, the virtual machine is invisible and unreachable from the outside internet; you cannot run a server this way unless you set up port forwarding (described below).
BRIDGE EXPLANATION
With bridged networking, VirtualBox uses a device driver on your host system that filters data from your physical network adapter. This driver is therefore called a “net filter” driver. This allows VirtualBox to intercept data from the physical network and inject data into it, effectively creating a new network interface in software. When a guest is using such a new software interface, it looks to the host system as though the guest were physically connected to the interface using a network cable: the host can send data to the guest through that interface and receive data from it. This means that you can set up routing or bridging between the guest and the rest of your network.
Source: networking – Can’t get dhcp ip in my virtual machine on Virtual box – Super User