tl;dr

LocalDate.parse( 
    "22/01/2010" , 
    DateTimeFormatter.ofPattern( "dd/MM/uuuu" ) 
)

…more…

// String input is:
// (a) long: "22/01/2010 to 23/01/2010". 
// (b) short: "22/01/2010".
// (c) null.

DateTimeFormatter f = DateTimeFormatter.ofPattern( "dd/MM/uuuu" ) ;

if( input.length() == 24 ) {           // Ex: "22/01/2010 to 23/01/2010"
    List<LocalDate> lds = new ArrayList<>( 2 );
    String[] inputs = input.split( " to " );
    for( String nthInput : inputs ) {
        LocalDate ld = LocalDate.parse( nthInput , f ) ;
        lds.add( ld );
    }
    … // Do what you want with `LocalDate` objects collection.

} else if( input.length() == 10 ) {    // Ex: "22/01/2010"
    LocalDate ld = LocalDate.parse( input , f ) ;
    … // Do what you want with `LocalDate` object.

} else if( null == input ) {
    … // Decide what you want to do for null input.

} else {
    System.out.println( "Unexpected input: " + input ) ;
}

See this code run live at IdeOne.com.

Using java.time

The other Answers use troublesome old date-time classes that are now legacy, supplanted by the java.time classes.

As for handling multiple types of strings, look at the length of the string.

if( input.length() == 10 ) { … }

If long, split on the 4-character substring “ to ”.

String[] inputs = "22/01/2010 to 23/01/2010".split( " to " );

Parse the date string as a LocalDate.

DateTimeFormatter f = DateTimeFormatter.ofPattern( "dd/MM/uuuu" );
LocalDate ld = LocalDate.parse( "22/01/2010" , f );

About java.time

The java.time framework is built into Java 8 and later. These classes supplant the troublesome old legacy date-time classes such as java.util.Date, Calendar, & SimpleDateFormat.

The Joda-Time project, now in maintenance mode, advises migration to the java.time classes.

To learn more, see the Oracle Tutorial. And search Stack Overflow for many examples and explanations. Specification is JSR 310.

You may exchange java.time objects directly with your database. Use a JDBC driver compliant with JDBC 4.2 or later. No need for strings, no need for java.sql.* classes.

Where to obtain the java.time classes?

• Java SE 8, Java SE 9, Java SE 10, and later
  • Built-in.
  • Part of the standard Java API with a bundled implementation.
  • Java 9 adds some minor features and fixes.
• Java SE 6 and Java SE 7
  • Much of the java.time functionality is back-ported to Java 6 & 7 in ThreeTen-Backport.
• Android
  • Later versions of Android bundle implementations of the java.time classes.
  • For earlier Android (<26), the ThreeTenABP project adapts ThreeTen-Backport (mentioned above). See How to use ThreeTenABP….

The ThreeTen-Extra project extends java.time with additional classes. This project is a proving ground for possible future additions to java.time. You may find some useful classes here such as Interval, YearWeek, YearQuarter, and more.

Use SimpleDateFormat:

SimpleDateFormat sdf = new SimpleDateFormat("dd/MM/yyyy");
String date = sdf.format("Your date");

Modern answer: Use LocalDate from java.time, the modern Java date and time API, and its toString method:

    LocalDate date = LocalDate.of(2012, Month.DECEMBER, 1); // get from somewhere
    String formattedDate = date.toString();
    System.out.println(formattedDate);

This prints

2012-12-01

Some will prefer the more explicit:

    String formattedDate = date.format(DateTimeFormatter.ISO_LOCAL_DATE);

ISO in the formatter name refers to the ISO 8601 standard. According to it a date is formatted the way you also want, as yyyy-mm-dd.

A date (whether we’re talking java.util.Date or java.time.LocalDate) doesn’t have a format in it. All it’s got is a toString method that produces some format, and you cannot change the toString method. Fortunately, LocalDate.toString produces exactly the format you asked for.

The Date class is long outdated, and the SimpleDateFormat class that you tried to use, is notoriously troublesome. I recommend you forget about those classes and use java.time instead. The modern API is so much nicer to work with.

Except: it happens that you get a Date from a legacy API that you cannot change or don’t want to change just now. The best thing you can do with it is convert it to java.time.Instant and do any further operations from there:

    Date oldfashoinedDate = // get from somewhere
    LocalDate date = oldfashoinedDate.toInstant()
            .atZone(ZoneId.of("Asia/Beirut"))
            .toLocalDate();

Please substitute your desired time zone if it didn’t happen to be Asia/Beirut. Then proceed as above.

Link: Oracle tutorial: Date Time, explaining how to use java.time.

SimpleDateFormat("MM-dd-yyyy");

instead of

SimpleDateFormat("mm-dd-yyyy");

because MM points Month, mm points minutes

SimpleDateFormat sm = new SimpleDateFormat("MM-dd-yyyy");
String strDate = sm.format(myDate);

String start_dt = "2011-01-31";

DateFormat parser = new SimpleDateFormat("yyyy-MM-dd"); 
Date date = (Date) parser.parse(start_dt);

DateFormat formatter = new SimpleDateFormat("MM-dd-yyyy"); 
System.out.println(formatter.format(date));

Prints: 01-31-2011

String dateStr = "Mon Jan 12 00:00:00 IST 2015";

First you have to parse your string to a date:

DateFormat parser = new SimpleDateFormat("E MMM dd HH:mm:ss Z yyyy");
Date date = parser.parse(dateStr);

and then you can format it:

DateFormat formatter = new SimpleDateFormat("dd.MM.yyyy");
System.out.println(formatter.format(date));