How to write this SQL query for a XML type?

What you want here is not entirely clear from your question. My guess is that you want the value from the <Value> node where the <Name> node is Tid.

Shred the XML using the nodes() function on /Information/Groups/Group/Items/Item and add a predicate [Name/text()="Tid"] to check the value for <Name>.

declare @T table(X xml not null);
insert into @T(X) values
('<Information>
  <Groups>
    <Group Name="Monitor">
      <Items>
        <Item>
          <Name>Cid</Name>
          <Value>2323232323</Value>
        </Item>
        <Item>
          <Name>Tid</Name>
          <Value>6656565656</Value>
        </Item>
      </Items>
    </Group>
  </Groups>
</Information>');

select I.X.value('(Value/text())[1]', 'bigint') as Value
from @T as T
  cross apply T.X.nodes('/Information/Groups/Group/Items/Item[Name/text()="Tid"]') as I(X);

If you only need <Name>Tid</Name>, your query would look something like this:

DECLARE @x XML = N'
<Information>
  <Groups>
    <Group Name="Monitor">
      <Items>
        <Item>
          <Name>Cid</Name>
          <Value>2323232323</Value>
        </Item>
        <Item>
          <Name>Tid</Name>
          <Value>6656565656</Value>
        </Item>
      </Items>
    </Group>
  </Groups>
</Information>
';

DECLARE @x_table TABLE (x XML);
INSERT @x_table ( x )
VALUES ( @x );

SELECT      ca.c.value('.', 'NVARCHAR(3)')
FROM        @x_table AS x
CROSS APPLY x.x.nodes('/Information/Groups/Group/Items/Item/Name/text()') AS ca(c)
WHERE       ca.c.exist('. [.="Tid"]') = 1;

If you need something else, let me know.