type
  TYourForm = class(TForm)
    ...
  private
    Canceled: boolean;
    ChunkSize: cardinal;
    procedure FileCopy(const sourceName, targetName: string; var canceled: boolean);
    procedure FilesCopy(const sourceFolder, targetFolder: string; var canceled: boolean);
  end;

procedure TDemoForm.FileCopy(const sourceName, TargetName: string);
var
  fsSource, fsTarget: TFileStream;
begin
  fsSource := TFileStream.Create(sourceName, fmOpenRead, fmShareDenyWrite);
  fsTarget := TFileStream.Create(targetName, fmCreate, fmShareDenyWrite);
  FileProgressBar.Max := fsSource.Size;
  Canceled := false;
  repeat
    FileProgressBar.Position := FileProgressBar.Position + fsTarget.CopyFrom(fsSource, chunkSize);
    Application.ProcessMessages;
  until Canceled or (FileProgressBar.Position = FileProgressBar.Max);
end;

procedure TDemoForm.FilesCopy(sourceFolder, targetFolder: string);
var
  sr: TSearchRec;
  fileSpec, sourceName, targetName: string;
  attr: integer;
begin
  attr := faArchive + faDirectory;
  sourceFolder := IncludeTrailingPathDelimiter(sourceFolder);
  targetFolder := IncludeTrailingPathDelimiter(targetFolder);
  if not DirectoryExists(targetFolder) then
    ForceDirectories(targetFolder);
  fileSpec := sourceFolder + '*.*';
  if FindFirst(fileSpec, attr, sr) = 0 then
  repeat
    if sr.Attr and faDirectory = 0
      then FileCopy(sourceFolder + sr.Name, targetFolder + sr.Name)
      else FilesCopy(sourceFolder + sr.Name, targetFolder + sr.Name);
  until Canceled or (FindNext(sr) <> 0);
  FindClose(sr);
end;