As Ray mentions, the xml methods are perfect for this.. here is an example based on your xml
DECLARE @xml XML
SET @xml = '<server name="Server1">
<log>
<logItem type="LogDate">
<value string="2012/12/15" />
</logItem>
<logItem type="error">
<value string="File not found" />
</logItem>
<logItem type="source">
<value string="Dir/SubDir/SomeFile" />
</logItem>
</log>
</server>'
SELECT @xml.value('(/server/log/logItem[@type="error"]/value/@string)[1]', 'varchar(50)')