Let say I have a dedicated server which is fully utilized by our website. Can I get another dedicated server from the same hosting company and have it both work together to provide double processing power for the website?
There are two options here, one would be to upgrade to a more powerful server if that is an option for you.
The second, which is what you specifically reference is known as load balancing. This is where you have two servers running side by side and where the traffic to your application is split between the two, in effect each server takes half of the traffic to your site. This can be done in one of two ways...
This is a hardware or software based solution depending on the provider whereby all traffic goes through one device and is then "balanced" between the end servers. This is the most intelligent method of doing it in the sense of machine intelligence as this can be configured to direct the user to the server with the least load at the time.
Multiple DNS A Records
This is the second way to achieve load balancing but does not assure even distribution of load across your services. With this method either server may be sent the traffic and once it has been connected to all connections from that network should go to the same server until the DNS cache expires and it hits your DNS server again.
In all honesty unless your website depends on high availability or another use case to justify the costs of a load balancer you may be better off to either upgrade your server or use DNS round-robin for load balancing.
There is always the option of upgrading/changing your server, or going for an n-tier solution, like @PlanetScale Networks describes in his answer.
One thing I would like to point out though is the benefits and money you can save, if before looking for more power and server resources, you try to optimize your applications and current systems.
In many cases, an un-optimized system/software is the cause of the excessive needs in resources. A single database query or a bad written script may consume all the available power/memory of the machine. Don't be surprised if you double the resources and are still seeing the server become overloaded.
So make sure your apps/sites/software are fully optimized, before you decide that you need a more powerful server.