You could try the following endpoint:
https://server/_api/Web/GetFolderByServerRelativeUrl(<folder url>)?$expand=Folders,Files
It returns files and folders beneath folder at the specified url.
JavaScript example
The following example demonstrates how to retrieve files and folder beneath 2015 folder:
var folderUrl = '/documents/2015';
var url = _spPageContextInfo.webServerRelativeUrl + "/_api/Web/GetFolderByServerRelativeUrl('" + folderUrl + "')?$expand=Folders,Files";
$.getJSON(url,function(data,status,xhr){
for(var i = 0; i < data.Files.length;i++){
console.log(data.Files[i].Name);
}
for(var i = 0; i < data.Folders.length;i++){
console.log(data.Folders[i].Name);
}
});
since IE9 you can also write:
$.getJSON(url,function(data,status,xhr){
data.Files.forEach(function( file ){
console.log( file.Name );
});
data.Folders.forEach(function( folder ){
console.log( folder.Name );
});
});
or if you need the individual filenames in an Array:
$.getJSON(url,function(data,status,xhr){
var filenamesArr = data.Files.map(function( file ){
return file.Name ;
});
var foldernamesArr = data.Folders.map(function( folder ){
return folder.Name ;
});
});