Check the request.setDestination functions here To Store file in External App-Specific Directory [example: "external/Android/data/your_app_name/filePath_you_set_in_function”], use like below:

DownloadManager.Request request = new DownloadManager.Request(uri);
request.setDescription("Selected Video is being downloaded");
request.allowScanningByMediaScanner();
request.setTitle("Downloading Video");
request.setNotificationVisibility(DownloadManager.Request.VISIBILITY_VISIBLE_NOTIFY_COMPLETED);
//Set the local destination for the downloaded file to a path within the application's external files directory
request.setDestinationInExternalFilesDir(context, Environment.DIRECTORY_DOWNLOADS, fileName); //To Store file in External Public Directory use "setDestinationInExternalPublicDir(Environment.DIRECTORY_DOWNLOADS, fileName)"
DownloadManager manager = (DownloadManager)
mContext.getSystemService(Context.DOWNLOAD_SERVICE);
manager.enqueue(request);

And if you want to download it to another place you need to move it after your download gets finished, by using IO streams. You can use a broadcast receiver to perform this task once the DownloadManager has finished downloading. You can use FileProvider to open the file with another app in Android version 10 or above.

The DownloadManager can only download to external storage, you need to implement your own download manager.

Use this method to save file:

// DownloadFile AsyncTask
    private class DownloadFile extends AsyncTask<String, Void, Bitmap> {

        @Override
        protected void onPreExecute() {
            super.onPreExecute();
        }

        @Override
        protected Bitmap doInBackground(String... URL) {

            String imageURL = URL[0];

            Bitmap bitmap = null;
            try {
                // Download Image from URL
                InputStream input = new java.net.URL(URL).openStream();
                // Decode Bitmap
                bitmap = BitmapFactory.decodeStream(input);
            } catch (Exception e) {
                e.printStackTrace();
            }
            return bitmap;
        }

        @Override
        protected void onPostExecute(Bitmap result) {

            if (result != null) {
                File dir = new File(mContext.getFilesDir(), "MyImages");
                if(!dir.exists()){
                    dir.mkdir();
                }
                File destination = new File(dir, "image.jpg");

                try {
                    destination.createNewFile();
                    ByteArrayOutputStream bos = new ByteArrayOutputStream();
                    result.compress(Bitmap.CompressFormat.PNG, 0 /*ignored for PNG*/, bos);
                    byte[] bitmapdata = bos.toByteArray();

                    FileOutputStream fos = new FileOutputStream(destination);
                    fos.write(bitmapdata);
                    fos.flush();
                    fos.close();
                    selectedFile = destination;
                } catch (IOException e) {
                    e.printStackTrace();
                }
            }
        }
    }

Usage:

new DownloadFile().execute("url_here");

Note: I have added code for image type file

Consider using Picasso for your purpose. I'm using it in one of my project. To save image on external disk you can use following:

 Picasso.with(mContext)
        .load(ImageUrl)
        .into(new Target() {
            @Override
            public void onBitmapLoaded(Bitmap bitmap, Picasso.LoadedFrom from) {
                try {
                    String root = Environment.getExternalStorageDirectory().toString();
                    File myDir = new File(root + "/yourDirectory");

                    if (!myDir.exists()) {
                        myDir.mkdirs();
                    }

                    String name = new Date().toString() + ".jpg";
                    myDir = new File(myDir, name);
                    FileOutputStream out = new FileOutputStream(myDir);
                    bitmap.compress(Bitmap.CompressFormat.JPEG, 90, out);

                    out.flush();
                    out.close();                        
                } catch(Exception e){
                    // some action
                }
            }

            @Override
            public void onBitmapFailed(Drawable errorDrawable) {
            }

            @Override
            public void onPrepareLoad(Drawable placeHolderDrawable) {
            }
        }
    );

From here you can download this library.

I can not believe i solved this. What I do is replacing:

getFilesDir()

to

Environment.getExternalStorageDirectory()

below I post my final code

    class DownloadFileFromURL extends AsyncTask<String, String, String> {
    ProgressDialog pd;
    String pathFolder = "";
    String pathFile = "";

    @Override
    protected void onPreExecute() {
        super.onPreExecute();
        pd = new ProgressDialog(DashboardActivity.this);
        pd.setTitle("Processing...");
        pd.setMessage("Please wait.");
        pd.setMax(100);
        pd.setProgressStyle(ProgressDialog.STYLE_HORIZONTAL);
        pd.setCancelable(true);
        pd.show();
    }

    @Override
    protected String doInBackground(String... f_url) {
        int count;

        try {
            pathFolder = Environment.getExternalStorageDirectory() + "/YourAppDataFolder";
            pathFile = pathFolder + "/yourappname.apk";
            File futureStudioIconFile = new File(pathFolder);
            if(!futureStudioIconFile.exists()){
                futureStudioIconFile.mkdirs();
            }

            URL url = new URL(f_url[0]);
            URLConnection connection = url.openConnection();
            connection.connect();

            // this will be useful so that you can show a tipical 0-100%
            // progress bar
            int lengthOfFile = connection.getContentLength();

            // download the file
            InputStream input = new BufferedInputStream(url.openStream());
            FileOutputStream output = new FileOutputStream(pathFile);

            byte data[] = new byte[1024]; //anybody know what 1024 means ?
            long total = 0;
            while ((count = input.read(data)) != -1) {
                total += count;
                // publishing the progress....
                // After this onProgressUpdate will be called
                publishProgress("" + (int) ((total * 100) / lengthOfFile));

                // writing data to file
                output.write(data, 0, count);
            }

            // flushing output
            output.flush();

            // closing streams
            output.close();
            input.close();


        } catch (Exception e) {
            Log.e("Error: ", e.getMessage());
        }

        return pathFile;
    }

    protected void onProgressUpdate(String... progress) {
        // setting progress percentage
        pd.setProgress(Integer.parseInt(progress[0]));
    }

    @Override
    protected void onPostExecute(String file_url) {
        if (pd!=null) {
            pd.dismiss();
        }
        StrictMode.VmPolicy.Builder builder = new StrictMode.VmPolicy.Builder();
        StrictMode.setVmPolicy(builder.build());
        Intent i = new Intent(Intent.ACTION_VIEW);

        i.setDataAndType(Uri.fromFile(new File(file_url)), "application/vnd.android.package-archive" );
        i.addFlags(Intent.FLAG_ACTIVITY_NEW_TASK);

        getApplicationContext().startActivity(i);
    }

}

Simply put this code to use this class

new DownloadFileFromURL().execute("http://www.yourwebsite.com/download/yourfile.apk");

This code can perform file download to your internal phone storage with a progress bar and continue asking your permission for application install.

Enjoy it.

If you are looking for saving image once and reusing it many times this may help you Google Volley Library. Universal Image Loader which is widely used after Volley.

EDIT : Now there are bunch of libaries which are so popular.

1. Picasso
2. Glide
3. Fresco

Choose according to your requirement.Here is comparison.

Try having a look at

_context.getFilesDir();

and

_context.getExternalFilesDir();

The problem is with your path to save the file , fileName shouldn't start with a / and apply this modification to your code .

File extStore = Environment.getExternalStorageDirectory();
File file = new File(extStore, fileName);

You should check media availability before trying to save .