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.
If you use Chrome, you can create multiple characters and use them side by side, each memory of its login. I use this regularly for several logs—Google and Microsoft servers. Chrome also has a MultiLogin extension that can do this. This allows you to subscribe to multiple accounts in just one browser window. I also work with Trello. The whole team uses it. But at the same time, our company uses all possible tools, for example, print screen tool. All this gives a good speed of completing the task and is of high quality.