Just to expand on what Tim said, what you are looking at is hosting a web application server. Generally, you will need the following:
- Web Application Server
- At least 1 public IP address
What you would do is either directly assign the public IP address to the Web Application Server (if you have more than 1 public IP address) or if you do not have any available (not currently used) public IP addresses then you create a Network Address Translation (NAT) rule on your firewall to redirect traffic sent to your firewall to your Web Application Server. For an example let's say your have a Web Application Firewall assigned a private IP address of 192.168.0.80. Your firewall has a public IP address of 18.104.22.168. The Web Application Firewall is running a web server on TCP port 80.
So you would make a NAT rule on the firewall that says inbound traffic on the external interface going to 22.214.171.124:80 is going to be translated on the internal interface to 192.168.1.80:80.
That is it as far as how it works. Before you go and do that though you need to make sure you have appropriate security in place. This includes (but certainly is not limited to) placing the Web Application Server in a Demilitarized Zone (DMZ), creating Access Control Lists (ACLs) that restrict only traffic to and from the Web Application Server to traffic that you should be permitted, an Intrusion Prevention System (IPS) to block bad traffic going to your Web Application Server, other typical security measures (e.g. patch management, anti-virus, anti-malware, OS hardening, etc.), and any special security requirements for your environment.
The task you are asking about is typical but not simple. You need to make sure you know how to securely do what your client is asking you before you do it so you don't expose their data to people that aren't supposed to access it.