The poisson distribution represents probabilities of counts of events. The lambda parameter is the mean expected number of events, but the actual random draws from the distribution (that is, the individual realizations of the poisson process) will be integers representing the number of counts actually observed.
For example, if we draw 100,000 values (100,000 individual realizations) for a poisson process with lambas of, respectively, 3, 3.4, and 3.95, the plots below show how many times we get 0 events, 1 event, 2 events, etc. for the 100,000 random draws. Note that all of the random draws are integers:
par(mfrow=c(3,1))
plot(table(rpois(1e5, 3)), xlim=c(0,15), main="lambda=3")
plot(table(rpois(1e5, 3.4)), xlim=c(0,15), main="lambda=3.4")
plot(table(rpois(1e5, 3.95)), xlim=c(0,15), main="lambda=3.95")