Step 1: Get the text from an element
WebElement element = driver.findElement([Idenitfy your element here]);
Once you get the element. Get content of the element
String sDate = element.getText();
if you are getting date value from Textbox, try
String sDate = element.getAttribute("value");
Step 2: Convert string to Date. e.g. Your date is February 3, 2015
DateFormat format = new SimpleDateFormat("MMMM d, yyyy", Locale.ENGLISH);
Date date = format.parse(sDate);
Step 3: Get today's date
Calendar c = new GregorianCalendar();
c.set(Calendar.HOUR_OF_DAY, 0); //anything 0 - 23
c.set(Calendar.MINUTE, 0);
c.set(Calendar.SECOND, 0);
Date today = c.getTime();
Step 4: compare two dates
boolean sameDay = date.get(Calendar.YEAR) == today.get(Calendar.YEAR) &&
date.get(Calendar.DAY_OF_YEAR) == today.get(Calendar.DAY_OF_YEAR);