If you are an admin you can adjust the order of the fields on the screen through the screen's field configuration (Go to Settings, Issues, Screens, and click "Configure" on the right for the screen you want to edit.) You can also add tabs with additional fields. However, beyond the order of the fields as they appear on the screen there is not much more you can do you change the UI.
Want to piggyback on Scott's post and let you know that for the most part, cloud UI isn't very configurable. If you're looking to update the position of fields that are standard, you won't have many options.
Can you provide any screen shots or mockups of what you'd like to see?
I believe Dung is referring to the NEW Issue UI which has been rolled out on Jira Cloud in Labs.
To answer your question, you can't change the order of the fields (they claim that is coming). They have completely done away with tabs. The only thing front and center is the description and comments all other fields show up in a skinny bar on the right in no apparent order.
The new UI is beyond awful. It appears to be designed by someone who doesn't actually use Jira. I strongly encourage you to provide feedback and open a ticket listing out your concerns.
Our Jira tickets now show up with a huge amount of white space and a TON of scrolling to see relevant information in the tiny side bar.
