By default Joomla!  'cleans' content and code on saving, removing code elements such as object, embed, video, audio, style and iframe etc..

This overrides code cleanup settings in JCE
(eg in Components >> JCE >> Global Configuration >> Cleanup HTML - make sure this is off first)

To fix this just do the following.

Joomla! 1.5

courtesy of - http://docs.joomla.org/Why_does_some_HTML_get_removed_from_articles_in_version_1.5.8%3F 

Why does some HTML get removed from articles in version 1.5.8?

In Joomla! 1.5.8 and later, a default filtering choice is in place for those who have not selected an Article Filter option. The default applied for those who have not made a selection implements "black list" filtering as a security precaution against possible XSS. In version 1.5.8, all users, including members of the Super Admin group, have this set by default. In versions 1.5.9 and later, users in the Super Admin group do not have any filtering by default.

Due to this change, you might notice problems trying to save a Youtube video, or embedding other Javascript or Flash objects into your Articles.

Changing the Article Global Configuration

If the default filtering option is not suitable for your needs, you can change it by using the Article Global Configuration options. For example, to change the default settings so that no filtering happens for members of the Super Administrator group:

1. Navigate to Content ? Article Manager.
2. Press the Parameters icon in the toolbar to show the Articles - Global Configuration screen.
3. Scroll down to Filtering Options, highlight all Filter groups exceptSuper Administrator, and select Filter type of Blacklist (Default), as shown in the screenshot below.
   

With this setting, Joomla! will use the default Blacklist filter for all users except for the Super Administrator group. If you want to let other groups of Users submit content with no filtering, just modify the instructions above to exclude these groups. If you want no filtering on any groups, follow the instructions above but select only the Registered group.

TinyMCE Editor Configuration

If you are using the Tiny MCE Editor, you may also want to adjust these settings in the plugin:

• Code Cleanup on Startup ? Off
• Code Cleanup on Save ? Always (TinyMCE since Joomla! version 1.5.12)
• Do Not Clean HTML Entities ? Yes (In Joomla! version 1.5.15 and later, it is called Entity Encoding -> Raw)

More detailed instructions for filtering options can be found by pressing the Help button when in the Article Manager, or here in the Joomla documentation.

Joomla! 1.6 & 1.7

1. In the Joomla! Article Manager, click on the Options button
2. In the Options Dialog, click on the Text Filters tab.
3. For the Super Users group, or any other trusted user group, set Filter Type to No Filtering. Make sure that you only set this option for user groups that can be trusted as setting the Filter Type to No Filtering will essentially allow the user to include any html in an article.

Joomla! 2.5

1. Go to Global Configuration
2. Click "Text Filters" Tab.
3. Change "Super Users" filter type to "no filtering"