Here is Microsoft's announcement of issues addressed in June's "Patch Tuesday". Three are rated "critical", Microsoft's highest rating. Three are rated "important" and one is rated "moderate". Many of the flaws patched have to do with remote code execution flaws and so are ones you will want fixed regardless of their actual rating.
This MS announcement is rather short and it looks like the company has exerted some effort to make them easier to read and digest, too, which is nice.