There are various different options available when selecting to create a new group on your site.
The most commonly used group is a User group. This is the yellow folder that has the most available options. This is the folder we'd recommend you use for the majority of groups.
Both the Features and Brochure groups have fewer options, and can be used for a brochure type site but if in doubt stick to creating user groups for all your content. You only have the option of having documents and media in a Brochure group, there is no ability to add members or events and therefore to create rotas and use the attendance and booking facility. Feature groups are similar to brochure groups but with the addition of Forums being available.
You can move articles between user, feature and brochure groups, but you won't be able to have sub groups of a different type within a folder type.
Directory groups are used for storing information on organisations and you can read more about these in the article about Organisation Directories
Payment groups/Payment pages are set up for groups you will use to take payments. You can read about setting up a payment processor and a payment group to take online donations here.
Mailing Lists are used for storing contacts, these are independent groups so you can set up group for users who are not site members, or you can just use them to store groups of people that you need to email frequently. You can add people into mailing lists from your root group, but if they are deleted from the root group they will remain in the mailing list because this is independent to the main database.
Independent groups are a separate database group in your site, you can have members in an independent group that don't appear in your main site root group. You can read more in the article about Independent Groups.