Next to a soldering iron a multimeter is probably one of the single most useful tools for working with electronics- it's an absolute must have item. A multimeter will allow you to check battery pack voltage, measure the amount of electrical current your servos or LEDs are using and allow you to check things like resistor values and test circuit continuity. In short, it's pretty darn hard to troubleshoot your electronic circuits without having a multimeter.
The most important features you need are:
AC/DC voltage measurement Current measurement Audible continuity test/diode test Resistance
Other features that are nice but not necessary:
Autoranging Capacitance Temperature Hold function
Like all test equipment you can spend a crazy amount of money but it's really not necessary. For working with microcontrollers, servos and LEDs you really just need an inexpensive basic meter. Both Adafruit and Sparkfun sell good meters for under $20. If you want a meter with more advanced features (the most useful of which is probably autoranging)Adafruit sells a nice Extech meter for $60. A decent quality meter will last you for many, many years. I've been using the same Wavetek Meterman multimeter for at least 15 years. I've dropped this thing countless times and it still works perfectly.
Dave of the EEVblog did a comparison of five $50 multimeters and tells you what features to look for-
Martin Lorton has a great series on how to properly use a multimeter-