'In the future' means at some unspecified time after now.
For example, "In the future, we might all live on Mars."
'In future' means 'from now onwards'.
For example, "I don't want this mistake to happen again. In future, I want to see copies of all invoices before payments are made."
I hope that makes sense. By the way, this distinction doesn't apply to US English. Speakers of American English don't distinguish between the two meanings, and use 'in the future' in both contexts.