Can I sign into two different user accounts on Trello at the same time using a single browser?
Clarification: I'd like to have them in two tabs on the same browser window.
If you use Chrome, you can create multiple personalities and use them side-by-side, each remembering its own login. I use this regularly for multiple Google and Microsoft cloud service logins.
Each personality has it's own set of history, cookies and credentials.
UPDATE: There is a Firefox plugin called Multifox that claims to allow multiple logins. Not tried it I'm afraid.
UPDATE2: Firefox has introduced a "Firefox Multi-Account Containers" capability via a plugin. However alternative containers are not included in the Firefox sync capability and it isn't clear how deep the isolation goes - cookies are mentioned but nothing else. Probably enough for Trello use though.
As mentioned in the comments, you can open one of them in a Private Window on Firefox, Incognito Window on Chrome or the equivalent on your browser. Alternatively you can open the accounts on completely separate browsers e.g Chrome and Firefox and then cascade the windows for the literal side-by-side.
Chrome has a extension called MultiLogin that can do this. It allows you to be signed into multiple accounts of just about anything within a single browser window.