Convert Date String to Date APEX

For converting a date string to date in APEX, Date.parse() and Date.valueOf is used. But to know the difference these two methods, accept only certain date string that can be parsed. 

Date.parse() - accepts date string in dd/MM/YYYY format.
Date.valueOf() - accepts date string in YYYY-MM-dd format.

In this case, if you have any specified format of date that is shown in the visualforce page, but want to perform a DML operation:

Make sure split of the date received and use the Date.newInstance and construct the date before parsing and performing DML.

Date mydate=Date.newInstance(integer.valueOf(year),integer.valueOf(month),integer.valueOf(day));

And in case if you don't know the format of the date string, use the below snippet to parse the date

Date theDate;
try {
  // parse works in the case of 10/11/1983
  theDate = Date.parse(dateString);
}
catch(Exception e) {}
if (theDate == null) {
  try {
    // valueof works in the case of 1983-10-11
    theDate = Date.valueOf(dateString);
  }
  catch(Exception e) {}
}
if (theDate == null) {
  // couldn't parse
}