How can I tell if I'm logged into my Dropbox account on the website or the app?

Hi there Bella Wizbang - thanks for using Dropbox and welcome to the Community!
 

You can check if you're logged into a Dropbox account account on the website by taking a look at the upper right corner. If you see your account's profile picture, it means you're signed in. 

On the desktop app, you can check the app's preferences under the Account tab and on the mobile app also navigate to the Account tab. 

I hope this helps!