How your web browser gets the IP address of a website?

Domain Name System (DNS) in the Internet that works tirelessly to give back the answer to your browser when it needs an IP address of a website. DNS is a hierarchical architecture that allows the resolution of the human readable names of a computer in the internet into a machine usable IP address. 

The process of resolving the website address, for instance, Youtube server, typed on the browser is as follows:

• As this is the first time accessing the www.youtube.com, the browser creates a DNS query to the local DNS server that’s configured in the operating system. This would be the DNS server in your ISP network (e.g. Comcast) which is called as caching DNS

• If the Comcast DNS never communicated with the Youtube server (which would be very unlikely), this wouldn’t find any entry in its cache. This DNS server makes a request to the root name server. The root name server is configured manually in the Comcast DNS server

• The root name server is the authoritative server that looks up the right most portion of the domain name and returns the name server(s) for the Top Level Domain (TLD) of the dot com (".com") domain

• As the Comcast DNS gets the "com" TLD name server, it makes another request to the "com" TLD DNS server which in turn returns the name of the Youtube's ISP (I presume it's Google) DNS server’s IP address.

• Now the Comcast's DNS server makes the last request to the Google's DNS server to resolve the IP address for the domain www.youtube.com. This ends the recursive calls made by this caching server and creates a cached entry of this resolved IP address with a Time To Live (TTL) value. The TTL tells when this cache entry would expire. It respond back to your computer (e.g. laptop) which made the initial request 

• Your laptop now knows the IP address to make the request to the www.youtube.com server to retrieve the web page on the browser. It would also cache the resolved entry in the Operating System level to make the subsequent request to the same server much faster by avoiding all the above calls until the cache entry expires

It's good to know the concept of DNS caching in a little detail.  DNS caching is the process through which the local DNS server (known as caching DNS) stores the already resolved IP address for a certain period of time. As mentioned in the above answer (5a), the computer operating system and local DNS servers first look into the stored cache for the IP address of a domain name. If the cache doesn’t have the record available, it reaches out to the authoritative DNS servers (root DNS, TLD DNS, destination  network’s DNS, etc.) to resolve for the IP and then stores that into the the caching DNS server as well as in the local computer’s OS. 

The caching of each record has an expiration time tagged along with that. This is called Time-To-Live (TTL) which is set (in terms of seconds) by the authoritative DNS. The cache is deleted when the TTL time is elapsed. At that time, if that domain name is requested for the IP address, it has to go through the same DNS resolution process. The DNS caching is used to improve the performance of the DNS resolution. This not only helps improving the performance of the network usage but also keeps the Internet free of DNS query traffics