The correct form is "today's mood," with an apostrophe before the "s."
The apostrophe in "today's" indicates possession. It shows that the mood being referred to belongs to or is associated with "today." In this case, "today" is acting as a noun, and the apostrophe followed by an "s" (today's) is used to create the possessive form.
For example:
- "Today's mood is joyful."
- "I can't explain today's mood; it's quite strange."
On the other hand, "todays mood" without an apostrophe would be considered incorrect because it lacks the possessive form, and it would not convey the relationship between the mood and the specific day "today."