This really is a good question which I have asked many times before. Not being a hardcore web designer (I'm more on the development side) I've usually asked the designers I work with about their opinions, and here is what I have distilled:
Using percentages vs pixels for sizing elements and so on is really a matter of personal taste or the requirements in each specific case. If you need it to scale, or it will scale well, use percentages. If not, use pixels. People here have used the examples of major columns in a page maybe needing to be fluid but borders may well need the precision of a pixel measurement.
Obviously sizing images is pretty much tied to their resolution and pixel units, so I always use them in this case.
However, using em sizing versus pixel or even point sizing for text is a whole different kettle of fish. Most of the guys I work with have a base reset style for font sizes which brings the size of 1em down to about 12px. They then use ems everywhere else (or as near to everywhere else as is possible) to size text elements, form controls and the like. This is how I operate now too as it seems to work well across a variety of browsers, OSes and DPIs on desktops and laptops. I can't vouch for mobile devices though.
Accessibility is the key though - if you need to make something usable for people with disabilities or work on a variety of devices out the box, even things you may consider old, then scaling will be a requirement. Build it into your model for designing the site from the get-go and you may realise that absolute pixel sizing is not even needed in that case.
For example, an art-heavy design is probably going to be designed at a single scale, but the new educational website for disabled users is going to have to work in a variety of situations.
Just remember, the W3C put a variety of methods for sizing and scaling into their specifications for a reason - flexibility. Do whatever fits best and works well for your audience (as Moustard said earlier).