Effective immediately, I am playing with the assumption that Laura Barton was Agent 19 and Bobbi's mentor in SHIELD, then passed the Agent 19 mantle onto Bobbi when she retired. They don't talk much anymore, because life gets in the way, but they're on good terms. 

Unless otherwise specified, I will also be assuming that MCU Clint was mentoresque with her (instead of their comics history). 

*those who don't partake of the MCU television programs.

Bobbi Morse (played by Adrianne Palicki in Agents of SHIELD) was until recently a SHIELD agent known as Agent 19 or Mockingbird. Trained initially as a biochemist, Bobbi eventually entered into the operations side of the agency, utilizing her abilities in combat (most notably with her trademark battle staves) and espionage (doing ops and going undercover as needed) and languages (she's fluent in several and conversational in still more). She was married to Lance Hunter for a short period and the marriage failed because of their differences, but they're still close in a strange way. After SHIELD fell to Hydra, Bobbi joined Robert Gonzales' "real SHIELD" alongside Isabelle Hartley, Alphonso Mackenzie, Anne Weaver, and others, but also worked for the properly reestablished SHIELD that was helmed by Phil Coulson, going undercover in Hydra for several months and ultimately rescuing Jemma Simmons in the process of escaping; the two quickly became close. Bobbi has also been close with fellow agents Daisy Johnson, Melinda May, Leo Fitz, Antoine Triplett (RIP), and others, and she seemed to have a history with Kara Lynn Palamas, whose capture by Hydra she inadvertently contributed to and was tortured for. In the course of what seemed to be a routine op with the team, Bobbi killed a Russian politician and she and Hunter were forced to exit SHIELD and presumably enter the real world as civilians. For threading's ease, the writer will either ignore this, pretend the banishment didn't last, or assume she's currently working as a solo vigilante hero. It's highly likely she knows most characters who've been tied to SHIELD, including at least some Avengers.

Bobbi has long been a character in Marvel comics and this writer has Done Her Research on Bobbi's comics history and uses it as needed, however incidentally (for example, Bobbi does wear glasses, although not every day, and the vigilante tendencies also count). The most established piece of this is that she was also once married to Clint Barton, aka Hawkeye; by extension, it's taken for granted that she knows the Avengers as she does in the comics, at least to some degree. Admittedly the writer also highly draws personality aspects from Chelsea Cain's eight-issue Mockingbird run because it was the most brilliant thing (sarcasm, cultural references, dominatrix tendencies, so-on).

Just a heads-up, if anyone is interested: Bobbi's high school/college/etc. self can now be found at mockingbirdies  for all of your aged-down adventures.

Short version:

• Kinsey 3 (approximately evenly bisexual) and aromantic.
• Kinky/toppy as fuck (yes to domming the hell out of partners in most agreed-upon contexts).
• As shipping goes: yes to Jemma, yes to Kara, yes to Isabelle, yes to Natasha, almost certainly yes to virtually any other MCU woman you can dream up, it's complicated but probably about Hunter and Clint, we can talk about other MCU film-canon guys, would entertain the possibility of Trip, almost certainly not about her other male teammates, get the fuck away Grant Ward.

( tl;dr )