// Get the reference to your ListView
ListView listResults = (ListView) findViewById(R.id.listResults);

// Set its adapter
listResults.setAdapter(adapter);

// Enable filtering in ListView
listResults.setTextFilterEnabled(true);

// Prepare your adapter for filtering    
adapter.setFilterQueryProvider(new FilterQueryProvider() {
    @Override
    public Cursor runQuery(CharSequence constraint) {

        // in real life, do something more secure than concatenation
        // but it will depend on your schema
        // This is the query that will run on filtering
        String query = "SELECT _ID as _id, name FROM MYTABLE "
                       + "where name like '%" + constraint + "%' "
                       + "ORDER BY NAME ASC";
        return db.rawQuery(query, null);
     }
});

Let’s say your query will run every time the user types in an EditText:

EditText queryText = (EditText) findViewById(R.id.textQuery);
    queryText.addTextChangedListener(new TextWatcher() {
        @Override
        public void beforeTextChanged(final CharSequence s, final int start, final int count, final int after) {

        }

        @Override
        public void onTextChanged(final CharSequence s, final int start, final int before, final int count) {
            // This is the filter in action
            adapter.getFilter().filter(s.toString());
            // Don't forget to notify the adapter
            adapter.notifyDataSetChanged();
        }

        @Override
        public void afterTextChanged(final Editable s) {

        }
    });