I think the compiler is pretty clear here: Your GetMimeType method is a static method, but the _mappings variable is not declared static (an non-static or instance field/variable).
If you want to use the mappings variable as it appears above do this:
private static IDictionary<string, string> _mappings = new Dictionary<string, string>(StringComparer.InvariantCultureIgnoreCase)
Edit: As the commenter pointed out below, you must be careful that this is actually the behavior you want. A static member means all instances will share this same mappings variable and can overwrite the data present. If you want one mappings variable per class, then you should change your method to an instance method (by removing the static keyword), as noted in the answer above.