Here's one important thing to know in addition to everything said before.
Query plans are often too complex to be represented by the built-in XML column type which has a limitation of 127 levels of nested elements. That is one of the reasons why sys.dm_exec_query_plan may return NULL or even throw an error in earlier MS SQL versions, so generally it's safer to use sys.dm_exec_text_query_plan instead. The latter also has a useful bonus feature of selecting a plan for a particular statement rather than the whole batch. Here's how you use it to view plans for currently running statements:
SELECT p.query_plan
FROM sys.dm_exec_requests AS r
OUTER APPLY sys.dm_exec_text_query_plan(
r.plan_handle,
r.statement_start_offset,
r.statement_end_offset) AS p
The text column in the resulting table is however not very handy compared to an XML column. To be able to click on the result to be opened in a separate tab as a diagram, without having to save its contents to a file, you can use a little trick (remember you cannot just use CAST(... AS XML)), although this will only work for a single row:
SELECT Tag = 1, Parent = NULL, [ShowPlanXML!1!!XMLTEXT] = query_plan
FROM sys.dm_exec_text_query_plan(
-- set these variables or copy values
-- from the results of the above query
@plan_handle,
@statement_start_offset,
@statement_end_offset)
FOR XML EXPLICIT