We know next to nothing about Beatrix prior to the events of the game. What little information we did have isn't even available in the game, but in the supplemental FFIX Ultimania book. At the time of Final Fantasy IX, Beatrix is 27 (soon to be 28). When she was 17, she was bested by Steiner (aged 23 at the time) in a personal contest. Unfortunately, we have no literally no further information on this (what a tease!). It would be very interesting to know if that fight has anything to do with the eyepatch she wears. After all, by the time of the events of the game, Beatrix has long been established as the best warrior in the land and has quite a fearsome reputation. If someone had bested her in recent times, it would have been mentioned.
Beatrix is first met in a brief scene near the very start of the game when Princess Garnet is attempting to escape Alexandria. She is the one who informs Queen Brahne that the Princess has taken the Royal Pendant - a stone precious to Alexandria - and appears to be missing. The Queen orders both Beatrix and Steiner to locate the Princess. From first impressions, Beatrix appears as a competent woman who is calm in an emergency and not a person who has time for pointless fuss. She is, in a word, professional - although she also comes across as a little imperious. Beatrix is someone secure in her skills and place in the world.
All in all, this scene is brief and a whole lot happens between then and the next time you see Beatrix. You'd honestly be forgiven for having forgotten about her at this point, but after this next encounter, you definitely won't forget her again. This confrontation happens in rainy Burmecia near the end of Alexandria's successful attack. Countless Burmecians have been killed, while many others have fled to Lindblum. The King is missing, and Freya - a Burmecia native and travelling companion of Zidane - fears the worst. They make it to the palace where they find Queen Brahne accompanied by her general, Beatrix. Although Freya has never seen her before, Beatrix is someone she knows by reputation.
Freya's long lost love, Sir Fratley, left Burmecia years ago to improve his skills and ability to defend his kingdom by seeking out renowned warriors throughout the land and battling them. He mentions Beatrix by name.
Sir Fratley: "I hear there are many fierce warriors out in the world - some more powerful than even I... ...Beatrix of Alexandria, in particular. They say her swordsmanship is the best in the land."
Sir Fratley is not the only one to have heard of Beatrix's abilities. There are a couple of other references throughout the game of her cold-bloodedness and powerful skills.
She also seems to invoke fear in those who know her - Thorn and Zorn, the court jesters who are in league with Kuja - declare that "very scary, it is, when the general gets mad." That could just be her reaction to Thorn and Zorn, though - they are very annoying. The Knights of Pluto are also intimidated by her.
While Freya is reminiscing about Sir Fratley, Queen Brahne and Beatrix are discussing their next move. The King of Burmecia has not been found, despite Beatrix's efforts. Kuja, the big bad, makes his first extended appearance and suggests that the Burmecians will have fled to the nearby Cleyra - a kingdom that has remained isolated for a long time behind a large sandstorm. While this gives Zidane (and the Alexandrians) their next destination, a lone Burmecian Soldier appears. No doubt running on grief and adrenaline, he challenges the Alexandrians.
Soldier: "Burmecia will never fall! Prepare to die!" Beatrix: "You wish to fight me, Beatrix of Alexandria?"
Upon hearing just who he has engaged, the soldier visibly wilts. He clearly knows who she is and is now having the biggest "I've made a huge mistake" moment of his life. The game is really building her up to have a chilling reputation.
Beatrix: "I commend your courage, but I will show you no mercy."
Luckily for this chap, Zidane, Freya and their companions arrive, which diverts attention away from the soldier and allows him to escape. Unfortunately for Zidane and Freya, this means her wrath is now focused on them.
Beatrix: "I once killed a hundred knights single-handedly... To me, you two are nothing more than insects."
Beatrix sounds arrogant, to be sure. But she can back up that arrogance by killing everybody in the room without breaking a sweat, so personally, I wouldn't be calling out this character fault to her face anytime soon. What's more, Beatrix is stone cold. She has undoubtedly killed many Burmecians on this day, and she will mercilessly cut down a lone soldier who is obviously no match for her. She is certainly a capable general, and is clearly a true soldier herself - she follows the order of her Queen without question and will take down anyone who gets in her way without hesitation or mercy. Her loyalty to her Queen may be admirable, but it's also hugely misguided. Perhaps, much like Steiner was, her loyalty made her blind to the twisted turn the Queen has taken.
Regardless of all this, right now this General "felled 100 men" Beatrix is a very immediate problem to Zidane and his friends. What follows is the first of three battles against her. Gird your loins, people.