You usually talk about your hopes for the future with the phrase "I hope ___":
I hope it doesn't rain today.
But sometimes English speakers say "I'm hoping ___" instead. "I'm hoping ___" expresses a little more sense of actively waiting to find out the answer. For example, you might say this if you're trying to sell your house:
I'm hoping that we'll be able to sell it by the end of next month.
You can also say "I'm hoping to ___":
I'm hoping to finish the last chapter over the weekend.