Indeed, saying "web-application", we usually mean a special application, programmed to run on the web-server all the time, just waiting for requests from user to process.
In your case, you have a console-based application.
Depending on the configuration of the server, no of these applications could be run on your client web-hosting, any of them or both.
Since, usually web-hosting is provided by hosting company, they may have configurations ready for running your applications, may have it turnable on/off or even take money for this.
In case of internal company server, you need to ask your customer and its IT-stuff to configure this.
Finally, you'll need to ask:
1. Does server support SSH? - it's simply a remote console. Usually it's running at port 22 and you many check it with command "telnet yourserver 22" (windows and linux) - if it doesn't reject your connect - means SSH is configured.
2. Does your server have java installed and if it is available for your account via SSH connection?
- Only if your customer really mean web-application instead of console based application you need to ask if server has web-application server for Java - usually, it's something like Apache Tomcat, Jetty, JBoss, Weblogic, etc. But this way will require application modification in order to run it in web-server.
If you will decide to use console-application and not "upgrade" it to web-application, you really may run it at the host your database is running (again, you'll need SSH). You'll save time on remote database access operations - theoretically, your program will work faster.