Can’t connect to a Vagrant/VirtualBox virtual machine

It's old but as there is no answer, I'll provide one. Command:

vagrant ssh

Is the equivalent of

ssh vagrant@localhost -p 2222 -i .vagrant/machines/default/virtualbox/private_key

This is default behavior, if you changed something change command appropriately. First of all Vagrant will create vagrant user on your guest box, and you will use that user to ssh. As previous folks said, it will forward traffic from port 2222 on your host to port 22 on your guest, by default (when you use vagrant up, you see that message being displayed). And lastly Vagrant creates keys for ssh session so you don't have to, so you need to provide public key as argument when connection via ssh.

You can work around this issue in one of two ways, both of which are in the VirtualBox manual:

Enabling DNS proxy in NAT mode

The NAT engine by default offers the same DNS servers to the guest that are configured on the host. In some scenarios, it can be desirable to hide the DNS server IPs from the guest, for example when this information can change on the host due to expiring DHCP leases. In this case, you can tell the NAT engine to act as DNS proxy using the following command:

VBoxManage modifyvm "VM name" --natdnsproxy1 on

Using the host's resolver as a DNS proxy in NAT mode

For resolving network names, the DHCP server of the NAT engine offers a list of registered DNS servers of the host. If for some reason you need to hide this DNS server list and use the host's resolver settings, thereby forcing the VirtualBox NAT engine to intercept DNS requests and forward them to host's resolver, use the following command:

VBoxManage modifyvm "VM name" --natdnshostresolver1 on

Note that this setting is similar to the DNS proxy mode, however whereas the proxy mode just forwards DNS requests to the appropriate servers, the resolver mode will interpret the DNS requests and use the host's DNS API to query the information and return it to the guest.