What exactly can someone do if they know your Wifi password?
A lot of things. Depending on how your network is configured, it could be like having an attacker having wired access to an Ethernet LAN. If your network does not allow client to client traffic, the attacker usually can talk only to the router and won't receive traffic from other computers. If the router does not allow administration access from the wireless network, the attacker can do pretty little on the network.
Can they only see which sites you visit or can they do more?
It depends on your network. Usually an attacker will not even be able to see the sites you access. Even if client-to-client is enabled (usually is) and the attacker uses ARP-spoofing to redirect all IP traffic to his computer, most sites today will use TLS and he will not be able to see what you are seeing. DNS traffic is in clear, and the DNS requests could be viewed by the attacker.
Are they able to hack(router) and do malicious things with your router just by knowing the password?
They can infer the router make and model, and depending on the configuration of the router they can attack the router. If the router have a default password, or vulnerabilities allowing unauthenticated command execution, they can change router settings. You could protect your router by not allowing connection from the wireless interface, only from the wired interface. This is a setting almost every wireless router have but are disabled by default on the majority of the routers I used.
Is it possible to send a virus through the wifi to the router?
Possible? Yes. But not needed. A hacker that have access to the router via wifi will be able to bruteforce the login. Depending on your password, they can access the administration panels without a virus.
Also can a hacker tell what kind of router you are using just by knowing your wifi password and loging into your wifi?
It depends on your settings, but generally they can. The router must tell every client the default gateway and network mask by using DHCP, and the attacker can use this information to probe the router. If wireless access is enabled, the attacker can download the login page, send incorrect data, and identify the router by the messages returned.