It's very possible (and no worries, basic questions are absolutely welcome).
What you probably want is an APIFunction deployed to the Wolfram Cloud. You can sign up for a (free) account on https://www.wolframcloud.com/, create a new notebook, and evaluate the following (copy & paste the code and press Shift+Enter):
CloudDeploy[
APIFunction[{"x" -> "Integer"}, #x + 1 &, "JSON"], "myapi",
Permissions -> "Public"]
This will create a cloud object (basically a file in the Wolfram Cloud) called "myapi" inside your personal cloud root directory. It's an API that takes an integer parameter x, increments it by 1 and returns the result as JSON (which is just a number in this case). It's public, so anyone who knows the URL can execute it. Example:
https://www.wolframcloud.com/obj/jpoeschko/myapi?x=5
You can read more about APIFunction, CloudDeploy, etc. in the Wolfram Language documentation. There are lots of examples and also guides such as how to Deploy a Web API.
APIFunctions consume Cloud Credits depending on the time they need to execute. With a free Basic subscription, you get a limited number of Cloud Credits each month, which might be enough for your purposes. If you need more, you can upgrade to one of the paid plans.
A note about names (which, admittedly, can be confusing): There's Wolfram (the company) and we make both Wolfram|Alpha (the web application where you can enter natural-language queries for computation) and the Wolfram Cloud (a platform where you can work with notebooks similarly to Mathematica, just on the web, and that allows you to deploy web applications and APIs based on the Wolfram Language). I assume your question is about Wolfram Cloud, not Wolfram|Alpha.
Hope this helps.
