Ceramic or electrolytic. Ceramics are more cost-effective nowadays for that sort of thing.
Ceramic's are the cheapest, but they cap out at 100uf, and are scarce above 10. You must use ceramics for bypassing (the ~0.1uf cap you need next to each IC). Now that they're available in such large values, they're practical to use for filtering too - but some voltage regulators aren't stable with ceramic caps (this is specified in the datasheet). These are typically your go-to for low value capacitors.
Electrolytics are available in larger sizes and for higher voltages than ceramics, and if you need a large (high value) capacitor, they're often your only choice. They are polarized, and have inferior reliability compared to ceramics. Traditionally used for power supply filtering, these are typically your go-to for higher values.
Tantalum caps are smaller than an electrolytic of similar size, with lower ESR as well, but are usually much more expensive. Advanced ceramic caps are encroaching on the traditional use cases for tant's - in the past, tant's were commonly used for bypassing. They are still widely used for power supply filtering and bypassing in portable devices. When these fail, they can smoke, catch fire, or explode - though if the manufacturer recommendations are followed, this is a very uncommon event. Polarized - fails catastrophically if polarity reversed.
Mylar/etc - Non-polarized caps available in higher voltages than ceramics are. Bulky and expensive - to be used only when other types don't meet the requirements of your project.
Ceramic and electrolytic capacitor assortments are available on ebay for stunningly low prices: