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uBlock supports this - you can right click on any element and select "Block element", which then pops up an editor allowing you to refine the CSS selector to only get the bit you want removed.

AdBlock (getadblock.com), competitor to Adblock Plus, has a wizard for blocking elements. Right click on the page, AdBlock -> Block This Ad -> adjust the slider for element-level specificity.

That creates a custom filter that you can go edit if you want.

If you just want to remove elements from the page, rather than block connections to third-party servers, I recommend Stylish. You can just write custom CSS rules and make those elements display:none. Alternatively, write a user script that does the same.

ABP prevents unwanted crap from being downloaded.

Remove it Permanently does this, adding a submenu to right-click: http://rip.mozdev.org/