Removing a link from a linked list
Code is fine I think, although according to Robert C. Martin, generally, if you need to add a comment then you have failed to write readable code.

For example, this could be made a little easier to read and understand by extracting the block 

else //customer is not the first element in the list 


into a separate method.  Also the long tests such as 

if ((currentLink.surname).equals(surname) &amp;&amp; (currentLink.firstName).equals(firstName))
while (currentLink.next!=null &amp;&amp; !((currentLink.surname).equals(surname) &amp;&amp; ((currentLink.firstName).equals(firstName)))
if (currentLink.surname).equals(surname) &amp;&amp; (currentLink.firstName).equals(firstName))


...contain duplicated code, namely

(currentLink.surname).equals(surname) &amp;&amp; (currentLink.firstName).equals(firstName)


which would be better extracted into, say testForMatch()

In Summary


Long methods are frowned upon
Repeated code is hard to understand and harder to maintain
Extracting into well named methods beats good comments any day (comments can be left behind during updates and cause more confusion than is necessary with just code)