In youtube, There are just 360p and 720p options .(no 240p,480p and 1080p)

HTML5 is usually better than Flash (particularly security wise), just Youtube's implementation uses Media Source Extensions, which is available in Firefox but disabled by default currently (EDIT: in recent Firefox this should no longer be a issue (since version 44) - IF you have the appropiate codecs installed on your system):

• Why you can’t play 1080p or 480p YouTube HTML5 videos in Firefox anymore - Nov 5 2013
• no option for 1080p youtube - Jan 25 2015
• How to get 1080p in Youtube’s HTML5 player in Firefox (on Linux) - Jun 29 2015

To fix it, basically go to about:config and:

• ignore the warranty dragons
• set the media.mediasource.enabled & media.mediasource.webm.enabled entries to true

The YouTube HTML5 player enabling page should look at minimum like this:

^{The more blue things the merrier}

Then sit back and enjoy 1080p playback (if it doesn't need to buffer for half a hour due to broadband bandwidth like mine does :)

If you need to enable MSE & H264, you should be able to do so using the method suggested here in about:config:

• Create a new boolean key called media.mediasource.ignore_codecs and set it to 'True' - it may now work so check https://www.youtube.com/html5
• If it does not work yet, do the following:
  • Set media.mediasource.mp4.enabled to 'True'
  • Set to 'True' all the things begining with media.fragmented-mp4 - except for media.fragmented-mp4.use-blank-decoder which needs to be set to 'False':

All should then work (does for me with Firefox Beta AND ffmpeg available... for ffmpeg on 14.* releases try this question):

EDIT: In Firefox 43 at least, doing the above may break the HTML5 player. There seems to be other issues (atleast with Firefox's useragent), but anyway if it breaks try setting media.fragmented-mp4.gmp.enabled to false (source)

EDIT: again in Firefox 44 (with the default FF useragent) some issues may occur, though for me it shows a error about HTML5 videos when actually becuase videos have been blocked for other reasons (e.g. copyright). I am now forcing HTML5 with this addon Another edit: now doesn't seem necessary on fresh profiles, possibly due to https://www.mozilla.org/en-US/firefox/44.0/releasenotes/#new

the problem is with the YouTube HTML5 player, I had the same problem and fixed at by installing flash plug-in. then go to this link and choose to disable the html5 player if it's still the default.

Afterwards, it should look like this:

It can also be caused by the lack of H.264 codec support on Firefox.

Installing the FFmpeg plugin for GStreamer solved this problem for me:

sudo add-apt-repository ppa:mc3man/gstffmpeg-keep
sudo apt-get update
sudo apt-get install gstreamer0.10-ffmpeg