Allah says that pilgrimage to the House is a duty owed to Him by people who are able to find a way to it (3:97). The Prophet ﷺ said that Hajj is once, and whoever does more, it is voluntary. Whoever performs Hajj for Allah's sake, without obscenity or disobedience, returns as on the day his mother bore him.
1Who must perform it
Every Muslim who is adult, sane and free, who is able to make the journey (istita'ah): enough money for the journey and for his family while away, safe roads and health.
A woman needs a mahram with her, according to the majority, and she must not travel alone. Some scholars (Maliki, and Shafi'i in a view) allow travel in a trustworthy group.
It is obligatory once in a lifetime and at once, once the conditions are met, in the view of most scholars.
2Why it matters
It is one of the five pillars of Islam.
Aisha asked the Prophet ﷺ about jihad for women, and he said that their best jihad is an accepted Hajj.
Where the schools differ
The four Sunni schools agree on the main ruling above and differ in these points. No school is preferred here.
Doing it at once or delaying it. the Hanafi, Maliki and Hanbali schools say it should be done at once; the Shafi'i school says it may be delayed, though it is better to hurry.
Hajj on behalf of another (Hajj badal) is allowed for the one who cannot go because of old age or illness, by one who has performed his own Hajj.
In it are clear signs [such as] the standing place of Abraham. And whoever enters it shall be safe. And [due] to Allah from the people is a pilgrimage to the House - for whoever is able to find thereto a way. But whoever disbelieves - then indeed, Allah is free from need of the worlds.
This is general information, not a fatwa. For your own situation, ask a qualified scholar.
Fiqh questions
Hajj: who must perform it
Allah says that pilgrimage to the House is a duty owed to Him by people who are able to find a way to it (3:97). The Prophet ﷺ said that Hajj is once, and whoever does more, it is voluntary. Whoever performs Hajj for Allah's sake, without obscenity or disobedience, returns as on the day his mother bore him.
Is this a fatwa?
No. It is a summary of well-known rulings with their evidence. For your own case, ask a qualified scholar.
What if my school differs from this answer?
Follow the view of the school you follow and the scholar who teaches you. The page lists the differences of the four schools.