Would it be possible to have one device (no router) in a network and then open a wifi from that device?
Let's say your friend borrows you your laptop and you want to connect to the network with your phone, but you don't know the password. Could you install a service or something on the laptop to make a new wifi?
And how would that behave with the internal IP's?
A WiFi access point is really two things. One of them is the system that handles key exchange and transmission of ethernet frames. The other is the one which takes over IP-level matters, usually a DHCP server that allocates an IP address to any client. Home routers commonly combine both functionalities, but this is not necessary.
For instance, at my home, I have two WiFi access points (one for each level), each with its own SSID and password. However, they are both connected to a shared ethernet link. One of the routers is also a DHCP server, while the other does nothing at the IP level -- it is in "bridge" mode. Devices can connect (for WiFi) to either access point, but they all end up on a single LAN and talk to the unique DHCP server, that grants them an IP address.
In the situation you are describing, a similar trick is at least conceptually doable -- the laptop serves as custom WiFi access point and merely forwards everything to the LAN, so your phone would be able to talk to the normal DHCP server, and get an IP from it.
Alternatively, the laptop could try to "hide" the presence of the phone, and use NAT so that any access from the phone would masquerade as a connection from the laptop. In that case, the rest of the network, including the DHCP server, only sees the laptop. Microsoft tends to call such a setup "connection sharing".
The real difficulty here is making the laptop act as a WiFi access point and simultaneously connect as a client to the office WiFi. Most WiFi hardware won't allow that. However, with a secondary WiFi (e.g. an USB-based WiFi dongle) or with a wired link, such a setup is relatively easy (depending on involved operating systems and user savviness with computers).
(Of course this all begs the question of why you do not know the WiFi password. Especially since your laptop knows it and could simply show it to you. Your question looks a bit like a request to learn how to circumvent your company's security policies, which is very rarely a good idea.)
Windows Internet connection sharing will do this. It has been available since Windows XP I believe. It will create a second network over WiFi if you are connected to the upstream network via ethernet. The important thing is that you have two network interfaces.
