Age
The narrator tells us that M. Gardiner is “several years younger than Mrs. Bennet and Mrs. Phillips,” though she does appear to be older than her Bennet nieces. Given that Mrs. Bennet is likely around 40, this probably puts her in her 30’s. Since her eldest child is 8 years old, she must be at least 25.
Family and Situation
We know that Mrs. Gardiner is originally from Lambton, Derbyshire, though we know nothing about her family (except that they were provincial enough to live in Lambton). There is significant evidence to suggest that her family moved out of county as we’re told that she lived there “about ten or a dozen years ago, before her marriage,” and we’re told that “though Wickham had been little there since the death of Darcy's father, it was yet in his power to give her fresher intelligence of her former friends than she had been in the way of procuring."
Mrs. Gardiner is married to Mr. Edward Gardiner, making her the aunt of the Bennet girls. We are told that she is particularly close to Jane and Elizabeth and that “they frequently had been staying with her in town.” Mrs. Gardiner also has four children of her own and lives in Gracechurch Street in London.
Other Connections
Though Mrs. Gardiner is aware of the grandeur of the Darcy family, it is clear she did not know them personally. All that she knows about Fitzwilliam Darcy is by reputation and she doesn't know enough about George Wickham to resist his explanation of events with the Darcys. Given that she left Derbyshire 10-12 years ago, when Wickham would have been around 15, this says that either he wasn't causing problems for townspeople at that time or M. Gardiner's delicate maiden ears were prevented from hearing about them.
Appearance
We don’t know anything about M. Gardiner’s personal appearance, but Elizabeth thinks Darcy will mistake the Gardiners for “people of fashion,” suggesting that they’re at least well put together.
Character
The narrator -- speaking a bit from Lizzy’s perspective -- tells us that Aunt Gardiner “was an amiable, intelligent, elegant woman, and a great favourite with all her Longbourn nieces.” We know that when her family comes to Longbourn for Christmas, “The first part of Mrs. Gardiner's business on her arrival was to distribute her presents and describe the newest fashions. When this was done she had a less active part to play. It became her turn to listen.” Overall, this suggests she is kind, indulgent and very close to her nieces.
Mrs. Gardiner also seems quite loyal. The narrator says that “On being made acquainted with the present Mr. Darcy's treatment of [Wickham], she tried to remember some of that gentleman's reputed disposition when quite a lad which might agree with it, and was confident at last that she recollected having heard Mr. Fitzwilliam Darcy formerly spoken of as a very proud, ill-natured boy.” Since everyone in Derbyshire seems to think so well of Darcy, this is likely a reflection of Mrs. Gardiner’s affection for Wickham (or better yet, Lizzy, who likes Wickham), rather than actual fact.
Ultimately, it is Mrs. Gardiner who has the clearest perception and gives some of the best advice about Elizabeth and Darcy, suggesting that she’s very sharp and is close to Elizabeth.
