This is too long to explain in a comment. Below is how the bridged adapter works.
Yet, if you can ping from the guest debian that means some form of DHCP happened and you must have and IP (which is a different IP from the host machine). I'd check router configuration to see if you're not allowing/disallowing specific MACs and/or IPs from the usage of TCP/UDP protocols (that is a measure some people use on their routers).
Bridged adapter
A bridged adapter on VirtualBox simulates as if there was a bridge (well it is easier to think of this as a hub) on the cable connection to your (host) actual machine. Even if the connection is over wireless the simulation is similar.
For (almost) all purposes the guest (debian) and the router the host machine is connected to are connected directly. Have separate MAC, separate IP, they even give separate ARP requests.
router <---> host machine running VirtualBox
^
|
+-----> guest machine running Debian
This is done by sending packets with a different hardware address (you can configure the hardware address in the adapter).
You need to configure the guest system (Debian) to connect to the network as if it was a separate machine. For example, if you use a WPA connection from the host machine you need to configure WPA (including the username and password) in the guest machine.
Best Answer
You may try ping an internet IP address, if it's connected by domain name doesn't work, go to edit resolv.conf to add name server manually resolv.conf nameserver 192.168.1.1
If you set up DHCP at first time to start the interface, this file will be written by DHCP service to the above information.