Two conditions using OR in XPATH

and and or are allowed inside the condition: [here]. Or you may also use multiple paths in one XPath expression using the pipe sign.

//PeopleList/Row[c1] | //PeopleList/Row[c2]

you can use or / and inside [....]

Example:

//*[contains('abc') or contains('def') or text()='abcdef']

More info about operators: http://www.w3schools.com/xpath/xpath_operators.asp

I don't think you need an "or" here. You just need to translate both operands to lower-case, rather than only translating one of them.

As noted by Michael Kay, no or is necessary.

Simply use:

PeopleList/Row
  [contains(translate(@LastName, 
                     'ABCDEFGHIJKLMNOPQRSTUVWXYZ', 
                     'abcdefghijklmnopqrstuvwxyz'), '" 
+
           translate(txtSearch.value,
                     'ABCDEFGHIJKLMNOPQRSTUVWXYZ', 
                     'abcdefghijklmnopqrstuvwxyz')'" 

+ "')]");