Which family of rocks is most likely to contain fossils?

There are three main categories of rocks: igneous, metamorphic, and sedimentary. These categories correspond to how the rocks form. Igneous rocks are formed when magma cools and hardens. Granite is a common example of igneous rock. Metamorphic rocks form from other rock that is subjected to a new extreme environment. Commonly, they form when rocks get buried deep within Earth's crust and are exposed to extreme heat and pressure. The rock never melts, but the heat and pressure cause the minerals to change in such a way as the resulting rock is considerably different from the original. Marble is a common example of a metamorphic rock. Sedimentary rocks are formed from layers of sediment that are compacted and cemented together. These sediments can come from other rocks, dissolved minerals that are left behind after evaporation, or from organic material from living organisms. Sandstone and limestone are common examples of sedimentary rocks.

Because sedimentary rocks are subjected to the least extreme environments and form where living organisms are, they are the type of rock that is most likely to contain fossils. This doesn't mean that other rock types don't have any fossils, just that sedimentary rocks most commonly have fossils. For example, imagine a mudslide occurs underwater in a lake or ocean. Many plants and animals get buried in this mudslide. Over time, the sediments compact and cement and become sedimentary rock. Traces of those living organisms that had been trapped in the mudslide will very likely be found in that sedimentary rock. By organisms becoming buried in sediment they are able to be preserved as fossils. 

Let's take a journey with this rock through an example of the rock cycle. Imagine after millions of years that sedimentary rock is eventually buried very deep, subjected to extreme heat and temperature, and becomes a metamorphic rock. The extreme heat and temperature may result in the loss of some of the fossils because of the drastic changes the rock went through. Fast forward millions of years more, and this metamorphic rock gets subducted underneath another tectonic plate and melts into magma. The fossils are now lost forever, and when that magma eventually hardens and becomes igneous rock, there won't be any traces of the fossils to be found. So, the reason that sedimentary rocks are most likely to contain fossils is because the environment in which they form is most conducive to preserving evidence of life.