Classic / Traditional / Formal
This is a timeless style that has developed since the first weddings were photographed. (Back in the 1840s if you're curious!)
Things have moved on significantly but the essence of the style remains the same. The photographer will usually work their way through a scripted 'shot list' to ensure they have everything covered that the couple have requested, including all the key events of the day and the details. Plus posed images of the couple on their own and with their families.
They will give plenty of direction to set up the shots and position people.
Documentary / Photojournalistic / Reportage / Candid
These terms are all interchangeable and mean pretty much the same thing. The photographer will follow you around and capture all the events of the day as they unfold, not just the planned moments. They give no direction and don't pose people, preferring an informal approach. A good documentary photographer will blend into the background and photograph without you really knowing they're there, generally focusing on authentic, emotional moments.
True documentary photographers don't edit their images, but this can vary so be sure to ask if you're considering this style and having smudged mascara cleaned up in Photoshop is important to you!
Artistic / Fine Art
This is a style that is quite difficult to define as exactly what is considered fine art is subjective. However, it generally means that the photographer has a specific 'look' that they want to achieve before taking the photo, which might be achieved with posing, angles, lighting or editing. Sometimes these techniques can be employed without the couple noticing and sometimes the shot will need to be set up in advance. It really depends on what the photographer's artistic vision is for the shot.
Editorial / Contemporary / Modern / Dramatic
These are very stylised and posed with the intention of producing dramatic, high-impact pictures. These images require a lot of direction from the photographer and use many lighting techniques from the fashion industry.
You're unlikely to get an entire wedding shot in this way but some couples love this style for their portraits and group shots.
Lifestyle combines a little bit of all of the above. I think it's best described as 'refined documentary with a little artistic direction.' A lifestyle photographer will tell the story of your day, providing lots of reportage style candid images. They will anticipate 'moments' and get themselves in the right position to capture these moments in an artistic and considered way, taking advantage of the lighting, or features of the venue for example. They generally have a 'hands-off' approach but might offer a little direction if they can see it would turn a great shot into an amazing shot, or be much more flattering for the subjects.
Photos of the details (flowers, table settings, decorations, shoes, rings etc...) are also photographed with the same thought process, as well as any posed 'formal group' images the couple have requested although these generally won't be included unless specified. However, when these are requested, the aim of a lifestyle photographer is to make them look as natural as possible. This is also true of couple's portraits - they will give some direction, usually just to get a natural reaction from the couple and introduce some movement while shooting a series of images in quick succession.
For me, lifestyle provides me with the flexibility to easily adapt to my couples' requirements and provide beautiful images that show everything they've invested their time and money on, at it's very best. My couples receive a curated gallery telling their wedding day story with all the details, and images of themselves and their guests looking relaxed and fabulous but without the photography dominating the day or getting in the way of the celebrations.