Here is a snippet that helps create a folder in a custom list (not a document library) using the SharePoint REST API (and jQuery).

This is the main function – no need to change anything in this one:

function createFolderInList(arg) {
    var deferred = jQuery.Deferred();
    var folder = {
        "__metadata": { "type": "SP.ListItem" },
        "ContentTypeId": "0x0120",
        "Title": arg.folderName
    };
    var useGUID = arg.listId.charAt(0) === "{";
    jQuery.ajax({
        "url": _spPageContextInfo.webServerRelativeUrl + "/_api/web/lists/" + (useGUID ? "getById('" + arg.listId + "')" : "getByTitle('" + arg.listId + "')") + "/items",
        "type": "POST",
        "contentType": "application/json;odata=verbose",
        "data": JSON.stringify(folder),
        "headers": {
            "Accept": "application/json;odata=verbose",
            "X-RequestDigest": jQuery("#__REQUESTDIGEST").val()
        },
        "success": function (data) {
            // Successfully created folder, but needs to update FileLeafRef
            var updData = { "__metadata": { "type": data.d.__metadata.type }, "Title": arg.folderName.split("/").pop(), "FileLeafRef": arg.folderName };
            jQuery.ajax({
                "url": _spPageContextInfo.webServerRelativeUrl + "/_api/web/lists/" + (useGUID ? "getById('" + arg.listId + "')" : "getByTitle('" + arg.listId + "')") + "/items(" + data.d.Id + ")",
                "type": "POST",
                "data": JSON.stringify(updData),
                "headers": {
                    "IF-MATCH": "*",
                    "X-HTTP-Method": "MERGE",
                    "accept": "application/json;odata=verbose",
                    "content-type": "application/json;odata=verbose",
                    "X-RequestDigest": jQuery("#__REQUESTDIGEST").val(),
                },
                "success": function () {
                    // Done creating and renaming folder
                    deferred.resolve();
                },
                "error": function (err) {
                    deferred.reject(err);
                }
            });
        },
        "error": function (err) {
            deferred.reject(err);
        }
    });
    return deferred.promise();
}

This is how you call it from your code:

createFolderInList({
    "folderName": "Created from code",
    "listId": "MyTestList", // List GUID or display name - if you use GUID it must include the curly braces around the GUID like this: {c5c44b98-34f1-4ade-87d3-ed292eee0d84}
}).done(function () {
    alert("Folder created");
}).fail(function (err) {
    alert("Failed to create folder:\n" + JSON.stringify(err));
});

You can create a subfolder by specifying the folderName like this:

"folderName": "Created from code/subfolder 1"

Alexander