This questions has a huge scope, and is not really a suitable question as you'll need to go away and learn several different technologies, make some design choices and choose some tools. I'm going to give you a few pointers.
First you'll need to decide on your server environment, Tomcat uses a Java web container so could be a good choice, but it's going to depend on the server you're hosting on. Then you are going to need to write a program that you'll install on the server that will provide you with a HTTP interface to your MySQL database (your API).
If you want something that's going to be of minimal effort, start with something like Grails (Groovy on rails). Grails programmes can be written in (Java based dynamic language) Groovy, or in Java, Groovy is quite straight forward to learn. Grails will build you a framework from your domain objects (ie the objects that translate into the tables in your database) and provide you with basic CRUD functionality that can be accessed by RESTful HTTP calls (once the program is running on a live server). So data can be passed from the device by using a HTTP POST method, and grails will take care of handling the data and saving it into the database, the reverse for retrieving information is also true.
That should give you a basic overview, I'd suggest starting by downloading Tomcat, and get a virtual server running on your machine.
http://tomcat.apache.org/
And downloading Grails
http://grails.org/
and going through books and tutorials. There is loads online you can google so I won't list them here, but hopefully this should help you get started. Good luck :)