Best before dates are labelled on foods that would not normally pose a risk to human health if consumed after the stated date. As the name suggests, the food is likely to be 'best' before the given date. This does not mean that the food cannot be consumed after that date, and it's often the case that these foods can be consumed after the stated date. Use a common sense approach: if it looks and smells ok, use it. Remember, sometimes foods have alternative uses when they are past their 'best': soft tomatoes are not nice in a salad, but can be excellent for making fresh pasta sauces. We say waste not, want not.
'Use by' dates, on the other hand, should be taken very seriously. Use by dates are given to foods which, as the people that know these things say, "from a microbiological point of view are highly perishable and are therefore likely after a short period to constitute an immediate danger to human health". Think chicken - you don't want to eat chicken that is out of date as it could be very dangerous. Of course, you should also use common sense alongside referring to use by dates: if the food smells bad and otherwise seems a bit dodgy, pass on it. It could be the case that the food had not always been stored as it should have been, and is therefore dangerous even though it's within its use by date.
You should know which foods you need to be strict with by looking out for use by dates. If it says "use by", it means use by. If it says "best before", it means best before. Simple!