Selenium not finding element

Is this a timing issue? Is the element (or the whole page) AJAX-loaded? It's possible that it's not present on the page when you're trying to look for it, WebDriver is often "too fast".

To solve it, is either implicit or explicit wait.

The Implicit Wait way. Because of the implicit wait set, this will try to wait for the element to appear on the page if it is not present right away (which is the case of asynchronous requests) until it times out and throws as usual:

// Sooner, usually right after your driver instance is created.
driver.manage().timeouts().implicitlyWait(10, TimeUnit.SECONDS);

// Your method, unchanged.
@Test
public void Appointments() {
    ...
    driver.findElement(By.id("ctl00_Header1_liAppointmentDiary")).doSomethingWithIt();
    ...
}

The Explicit Wait way. This will only wait for this one element to be present on the page when looking for it. Using the ExpectedConditions class, you can wait for different things, too - the element to be visible, clickable etc.:

import static org.openqa.selenium.support.ui.ExpectedConditions.*;

@Test
public void Appointments() {
    ...
    WebDriverWait wait = new WebDriverWait(driver, 10);
    wait.until(presenceOfElementLocated(By.id("ctl00_Header1_liAppointmentDiary")))
        .doSomethingwithIt();
    ...
}

You're searching for ctl00_Header1_liAppointmentDiary, but there only is Header1_liAppointmentDiary, those are not the same...

ctl00_Header1_liAppointmentDiary != Header1_liAppointmentDiary

There is no element with id="ctl00_Header1_liAppointmentDiary" in your html

driver.findElement(By.id("ctl00_Header1_liAppointmentDiary"));

Should be

driver.findElement(By.id("Header1_liAppointmentDiary"));

try driver.findElement(By.ClassName("MyAppointments"));

If webdriver can't find an element by xpath or id, it is usually a good idea to try all the By options that could work

http://selenium.googlecode.com/svn/trunk/docs/api/java/org/openqa/selenium/By.html

Looking at the code I think the link you are trying to click is under a drop down or you need to mouse over something to see this link. If it is so then you will have make the element visible to perform the action.