Princess Eugenie of York is 8th in line to the throne, however, she isn’t expected to lead a life of royal engagements or public duties.
If she were to marry Jack Brooksbank, a few things would change.
Firstly, it would be unlikely (but not completely impossible) that Mr Brooksbank would be granted a title by Her Majesty, meaning he would remain a commoner.
If he was granted a title, it would likely be a non-royal peerage such as an Earldom, though this in itself is unlikely, as previously explained.
Because a woman takes her husband’s style and precedence unless hers is higher, (which it is), her precedence will not be affected.
She may choose, however, to follow past precedent and style herself as Her Royal Highness Princess Eugenie, Mrs Jack Brooksbank, or even just privately style herself as Mrs Brooksbank. Equally she may choose just to continue being Princess Eugenie.
Traditionally, female royals drop their territorial designation when they marry, so either way, she probably wouldn’t be referred to as Princess Eugenie of York any more (rather just Princess Eugenie).
It is highly unlikely that the wedding would be a public event like that of Princess Eugenie’s cousin, Prince William and his wife the Duchess of Cambridge, but likely to be a private family affair like that of Zara Phillips and Mike Tindall.
Because Princess Eugenie is eighth-in-line to the throne, The Queen does not need to give her consent for her marriage to take place. This is reserved for members of the Royal Family who are sixth-in-line to the throne or higher. Similarly, if her sister Princess Beatrice got married, she would no longer have to seek permission from Her Majesty as Princess Charlotte’s birth pushed her to seventh-in-line.