Yasukichi the Egg-Catcher Man, commonly referred to simply as The Egg-Catcher Man, is a traveller who used to live in Furinkan while Kasumi was still a toddler, where he lived and worked on a Chicken Hatchery. He eventually had to leave Kasumi due to his family moving, but not before she gives him a charm which she'd gotten from her grandmother.
Yasukichi has a very gentle personality and would rather run away from danger than fight it (as shown by how he desperately tries to run away from the Pit bull rather then quickly, and probably easily, defeat it). He is also very willing to put the feelings of others before his own personal feelings, the most prominent example being his quick acceptance that Dr. Tofu should be Kasumi's husband rather than him, despite the feelings he still harbours for her since childhood.
Yasukichi and Kasumi were small children together, playmates, friends and possibly a childhood crush of each other. Yasukichi's family ran a nearby chicken hatchery, which instilled in him the seeds of his future abilities. They were inseperable until the day that Yasukichi had to move away with his family; as a token to remember her by, Kasumi gave her friend a charm that she had received from her grandmother, promising it would bring him good luck. It was actually a ward for safe childbirth, but Yaskichi took it all the same.
One cold winter's evening, Yasukichi wandered back into Tokyo, having become a vagabond as he grew up. He had a chance encounter with Kasumi, returning with shopping for the evening meal, and impressed her by catching the contents of her carton of eggs when she tripped. The two of them chatted amicably for a moment, then went their seperate ways; while Kasumi's upbeat attitude would be interpretated by the other residents of the Tendo Dojo as her having been courted by the vagabond, the meeting was quite innocent.
Resuming his wanderings, Yasukichi was attacked by a bulldog, and ended up stumbling into Dr. Tofu's Clinic, where the kindly Tofu Ono treated his injuries, fed him and allowed him to stay for the night. Soon after, Yasukichi beheld what happened to his benefactor when he came into contact with Kasumi Tendo, and while initially angry that Tofu would become such a bumbling fool around a girl he liked, decided that he owed Tofu.
So, he set out to the Tendo Dojo, hoping to gently nudge Kasumi and the good doctor together. With a few innocent comments and a little advice, he sent Kasumi off to take some food to Dr. Tofu, then decided he had given them a sufficient push and headed on away from town. As he left, though, Kasumi finally saw the charm he wore around his neck and recognized him, but she chose not to say anything about who he was or call him back.
Yasukichi is an expert in all things related to eggs, able to accurately judge size, weight, palability and the dish it is best suited for with only a brief visual inspection.
Yasukichi also has impressive speed, reflexes and hand-eye coordination for a non-martial artist, routinely being able to pull off feats like snatching up a whole baker's dozen of eggs from midair and placing them in a basket without breaking or dropping a single one.
Yasukichi and Kasumi Tendo were childhood friends and one or both of them may have had a crush on the other. When they meet again, Kasumi doesn't recognize him at first and chooses not to make a fuss by calling out when she does recognize him again. It is unknown if they retain any romantic interest in each other -- even if they do, Yasukichi evidently believes that Kasumi deserves better then him, and would much rather gently nudge her towards Dr. Tofu then try to court her.
Yasukichi greatly respects Dr. Tofu and considers him a friend for his generosity in treating the vagabond for free. While he finds it annoying that Dr. Tofu can act like such a bumbling fool around Kasumi, he does consider the two to make a good match and would like for them to be married.
Yasukichi is politely indifferent towards the other residents of the Tendo Dojo. In return, Ranma Saotome is of the same attitude towards Yasukichi (though he takes Yasukichi's skill at catching eggs as a harmless personal challenge), while Akane is initially wary of him (believing he is seeking to steal away Kasumi, who "belongs with" Dr. Tofu), and Soun is hostile towards him for similar reasons to Akane.