Route53 (https://aws.amazon.com/route53/) refers to the TCP port 53 that's used for DNS servers, and is Amazon's DNS service. Similar to what you would do with BIND (http://www.isc.org/downloads/bind/), you can define DNS entries in Route53, and set up the service to automatically route the requests to specific AWS services that host applications or files.
DNS is a critical part of a deployment. It needs to be highly available, and to route each incoming request as fast as possible. If you are deploying your services on AWS, it is highly recommended to use Route53;or to use the DNS provider of the company where you bought the domain, and not deal with DNS yourself.
Route53 can work in close cooperation with Elastic Load Balancing (ELB) (https://aws.amazon.com/elasticloadbalancing/), which is a load balancer that can be configured to distribute incoming requests to several backends. Typically, if you are deploying several VMs for the same microservice...