First, to answer your specific questions: in your first example with
car accidents, you would indeed take m=3, which is the expected number
of fatal accidents in a year. As for x, you would set x=5 to get the
probability p(x;m) of exactly 5 fatal accidents occurring in a given
year in that intersection.
In the second example, m will be equal to 18, since if the expected
number of particle hits every 10 minutes is 3, then in one hour it
will be 3*6=18. x would be 30, since you are interested in the
probability of getting exactly 30 hits.
Now, for a more general explanation: The Poisson distribution
describes the number of "rare events" happening in a given time/space
interval. As you seem to already understand pretty well, the type of
"rare events" can be anything from car accidents, radioactive particle
disintegration, to rain drops hitting a sidewalk or spelling mistakes
in a newspaper page.
The function p(x;m) gives the probability that exactly x events will
happen, provided that m is the expected number of events happening. It
is given by the expression
p(x;m) = e^(-m) m^x / x! ("^" means "to the power of", "!" means
(x = 0,1,2,3, ...)
Sometimes you know the "m" for a different time or space interval than
that for which you want to compute probabilities. For instance, if I
know that an average page of a certain newspaper contains 5 spelling
mistakes, then the probability that on a given day there will be 7
mistakes on the first two pages will be p(7;10) - I take m=5*2 to
account for the number of pages I look at.
Poisson Distribution - from Eric Weisstein's world of Mathematics
Hope this helps,