How to implement my very own URI scheme on Android

This is very possible; you define the URI scheme in your AndroidManifest.xml, using the <data> element. You setup an intent filter with the <data> element filled out, and you'll be able to create your own scheme. (More on intent filters and intent resolution here.)

Here's a short example:

<activity android:name=".MyUriActivity">
    <intent-filter>
        <action android:name="android.intent.action.VIEW" />
        <category android:name="android.intent.category.DEFAULT" />
        <category android:name="android.intent.category.BROWSABLE" />
        <data android:scheme="myapp" android:host="path" />
    </intent-filter>
</activity>

As per how implicit intents work, you need to define at least one action and one category as well; here I picked VIEW as the action (though it could be anything), and made sure to add the DEFAULT category (as this is required for all implicit intents). Also notice how I added the category BROWSABLE - this is not necessary, but it will allow your URIs to be openable from the browser (a nifty feature).

Complementing the @DanielLew answer, to get the values of the parameteres you have to do this:

URI example: myapp://path/to/what/i/want?keyOne=valueOne&keyTwo=valueTwo

in your activity:

Intent intent = getIntent();
if (Intent.ACTION_VIEW.equals(intent.getAction())) {
  Uri uri = intent.getData();
  String valueOne = uri.getQueryParameter("keyOne");
  String valueTwo = uri.getQueryParameter("keyTwo");
}

I strongly recommend that you not define your own scheme. This goes against the web standards for URI schemes, which attempts to rigidly control those names for good reason -- to avoid name conflicts between different entities. Once you put a link to your scheme on a web site, you have put that little name into entire the entire Internet's namespace, and should be following those standards.

If you just want to be able to have a link to your own app, I recommend you follow the approach I described here:

How to register some URL namespace (myapp://app.start/) for accessing your program by calling a URL in browser in Android OS?