Sample for accessing a local database from JavaFX.

H2app.java

import java.sql.*;
import java.util.logging.*;
import javafx.application.Application;
import javafx.collections.*;
import javafx.event.ActionEvent;
import javafx.event.EventHandler;
import javafx.scene.Scene;
import javafx.scene.control.*;
import javafx.scene.layout.*;
import javafx.stage.Stage;

public class H2app extends Application {
  private static final Logger logger = Logger.getLogger(H2app.class.getName());
  private static final String[] SAMPLE_NAME_DATA = { "John", "Jill", "Jack", "Jerry" };

  public static void main(String[] args) { launch(args); }

  @Override public void start(Stage stage) {
    final ListView<String> nameView = new ListView();

    final Button fetchNames = new Button("Fetch names from the database");
    fetchNames.setOnAction(new EventHandler<ActionEvent>() {
      @Override public void handle(ActionEvent event) {
        fetchNamesFromDatabaseToListView(nameView);
      }
    });

    final Button clearNameList = new Button("Clear the name list");
    clearNameList.setOnAction(new EventHandler<ActionEvent>() {
      @Override public void handle(ActionEvent event) {
        nameView.getItems().clear();
      }
    });

    VBox layout = new VBox(10);
    layout.setStyle("-fx-background-color: cornsilk; -fx-padding: 15;");
    layout.getChildren().setAll(
      HBoxBuilder.create().spacing(10).children(
        fetchNames, 
        clearNameList    
      ).build(),      
      nameView
    );
    layout.setPrefHeight(200);

    stage.setScene(new Scene(layout));
    stage.show();
  }

  private void fetchNamesFromDatabaseToListView(ListView listView) {
    try (Connection con = getConnection()) {
      if (!schemaExists(con)) {
        createSchema(con);
        populateDatabase(con);
      }
      listView.setItems(fetchNames(con));
    } catch (SQLException | ClassNotFoundException ex) {
      logger.log(Level.SEVERE, null, ex);
    }
  }

  private Connection getConnection() throws ClassNotFoundException, SQLException {
    logger.info("Getting a database connection");
    Class.forName("org.h2.Driver");
    return DriverManager.getConnection("jdbc:h2:~/test", "sa", "");
  }

  private void createSchema(Connection con) throws SQLException {
    logger.info("Creating schema");
    Statement st = con.createStatement();
    String table = "create table employee(id integer, name varchar(64))";
    st.executeUpdate(table);
    logger.info("Created schema");
  }

  private void populateDatabase(Connection con) throws SQLException {
    logger.info("Populating database");      
    Statement st = con.createStatement();      
    int i = 1;
    for (String name: SAMPLE_NAME_DATA) {
      st.executeUpdate("insert into employee values(i,'" + name + "')");
      i++;
    }
    logger.info("Populated database");
  }

  private boolean schemaExists(Connection con) {
    logger.info("Checking for Schema existence");      
    try {
      Statement st = con.createStatement();      
      st.executeQuery("select count(*) from employee");
      logger.info("Schema exists");      
    } catch (SQLException ex) {
      logger.info("Existing DB not found will create a new one");
      return false;
    }

    return true;
  }

  private ObservableList<String> fetchNames(Connection con) throws SQLException {
    logger.info("Fetching names from database");
    ObservableList<String> names = FXCollections.observableArrayList();

    Statement st = con.createStatement();      
    ResultSet rs = st.executeQuery("select name from employee");
    while (rs.next()) {
      names.add(rs.getString("name"));
    }

    logger.info("Found " + names.size() + " names");

    return names;
  }
}

Answer to StackOverflow question: javafx connection to mysql.
Actually interfaces to the H2 database instead of MySQL (but interface to MySQL would be pretty similar).
Uses jdbc as the database interface (which is a good match for a simple app) - larger, more complex applications may benefit from use of a technology such as jpa.
Full (NetBeans) sample project is available as a zip file.
A nice modification to the sample would be to run Database queries as Tasks so that the JavaFX application thread is not blocked while the database is being accessed (for the trivial sample here it is not necessary, but for a larger database with more complex queries, such an implementation would be desirable).

Created an additional Task based implementation to demonstrate running the database operations on their own thread as requested by the StackOverflow question: JavaFX - Background Thread for SQL Query