There are different options to distribute network Traffic using Microsoft Azure. These options work differently from each other, have different feature sets, and support different scenarios. They can each be used in isolation or in combination.

Azure Load Balancer
Works at transport layer (level 4 of OSI network stack)
Provides network-level distribution of traffic across instances of the application running in the same Azure datacenter
Use Load Balancer to:
Balance incoming traffic to VMs from the Internet (Internet-facing load balancing)
Balance traffic between VMs in the VNets or cloud services (Internal Load Balancer)
Balance traffic between on-premises servers and VMs (Internal Load Balancer)
Forward traffic to specific VM

Configure Load Balancer Requirement

Distribution mode for Azure load balancer
Hash-based distribution mode
Source IP affinity mode / session affinity

Configure Load Balancer:
Port Forwarding
Automatic Reconfiguration
Service Monitoring to probe the health of the various server instances (see LoadBalancerProbe Schema)
Source NAT
Deploy Network Load Balancer: Azure Classic vs Azure Resource Manager

Traffic Manager
The most important point to understand is that Traffic Manager works at the DNS level.
Uses DNS responses to direct end-user traffic to globally distributed endpoints running in different datacenters around the world.
Clients connect directly to endpoints

Application Gateway
Works at application layer (level 7 of OSI model)
Works as a reverse-proxy service
Client connections are terminated at gateway and then forwarded to application