I want to connect powered studio monitors to a laptop. Do I need a mixer?

You need an external sound card, it plugs into a USB port and you plug in the jack from the monitors into the soundcard. Although some midi controllers such as Traktor S4 have built in soundcard.

1/8 splitter to to dual 1/4 or RCA or XLR (depending on the studio monitor) will work, but most of the time you'll want to go with an external sound card such as the Sapphire 6 USB (what I use), or one of the many offerings from M-Audio, PreSonus, and companies as such. The external sound card will greatly improve your sound quality and also let you (depending on the model), connect microphones and instruments if you wish to record music. The mixer option will also work with any mixer with a USB sound card built-in, but unless you plan on using the laptop primarily for DJ'ing or studio production, I would suggest an external sound card because mixers can take up quite a bit of space.

You can just uses the stereo mini jack output of your laptop if you have a splitter adaptor from minijack stereo plug to two mono plugs (either xlr or 1/4"). This will work but the audio quality is limited by the minijack output. Best to get a stereo USB or Firewire digital to audio convertor - then plug the speakers into its stero outs.

Assuming your studio monitors are RCA, you could just go with 1/8 to RCA cables.