“It's been one thing after another all day.”
You've had a busy day at work, with lots of different people asking you for help with different problems. Now it's the afternoon, and you're talking with a coworker. You want to tell her about your busy day, so you tell her this.
It's been one thing after another all day.
You use "has been" to describe a situation that started in the past and continued until now. The situation can be something that happens continuously:
They've been arguing with each other.
She's only been gone one day.
Or it can be an action that is repeated over and over for a length of time:
I've been going to that grocery store for over 25 years.
The phrase "It's been" is used to describe your general recent situation. It's like saying "my life has been ___":
It's been tough looking for a job in this economy.
It's been a fun vacation, but honestly I'm ready to get back to my routine.
Or you can use "it's been ___" to talk about some object. For example, if the coffee machine at your office doesn't work, you can describe it like this:
It's been broken for a few days.
"One thing after another" means a series of small problems or annoyances. You use it when problems keep happening again and again.
You can use it like this:
It's one thing after another with him!
It's been one thing after another all week!