Do I need to have a DNS server?

Yes. You cannot find websites without a DNS server. Is the Time Capsule used as your network router? If not then it doesn't need DNS servers. If it is the main router, then the default DNS server is the router's network address, such as 10.0.1.1 or 192.168.1.1. Or you can use any publicly accessible DNS such as OpenDNS - 208.67.220.220.

DNS service is typically provided by your ISP or dedicated DNS services, like OpenDNS or Google to name a few. Simply DNS is analogous to a phone book. It allows us to use a URL, like www.apple.com, instead of an IP address to access a website or server over the Internet.

By default, most consumer-grade Internet routers provide DHCP service, which is responsible to provide local network clients with the necessary IP addresses to access resources on the Internet. These also include the DNS server addresses. In turn, most network clients are also DHCP clients so that they can receive these addresses.

You can see those provided IP addresses on your MacBook Pro, by going to:

• System Preferences > Network > Wi-Fi or Ethernet (depending on how you connect your MBP to your local network)
• There you will find your device's assigned IP addresses for itself, your router and the DNS servers that have been provided it by the DHCP service running on your Time Capsule.

Is this image telling me..."device's assigned IP addresses for itself, your router and the DNS servers that have been provided it by the DHCP service running on your Time Capsule"?

That is correct.