The word rational is derived from the Latin word ration. It entered Old French as reisun or reson, then entered Middle English as resoun or reisun. This tells us that the English word rational is directly tied to the concept of reason as in reasoning powers, cognitive powers, sanity, logic, logical thinking. Therefore, a rational being is one who is capable of logical thought with the ability to reason toward sound conclusions based on facts and evidence, draw inferences from situations and circumstances, make sound well-reasoned judgements based on factual information. Humans--at their best--are rational beings who are unfortunately given to moments of irrationality during which decisions, actions, judgements, conclusions and inferences are based on emotion, persuasion, pressure, physicality, anything other than sound reason.
When the Latin prefix ir- is added to the word rational, the word takes on a negative force since ir- is a variation of the negating third usage of the Latin prefix in-. So an irrational being is one who does not posses--or utilize--the faculties of cognitive, logical, inductive and deductive thought that can be used to draw inferences and conclusions or make judgements. Animals are often said to be irrational as compared to humans (although the latest cat and dog research has modified this concept somewhat). Someone who is behaving without using their rational, reasoning mental faculties and powers (e.g., hyper-emotionally) is said to be behaving irrationally. Some psychological conditions such as paranoia render individuals irrational to varying degrees.
[These definitions and the etymology are drawn from Random House Dictionary as presented on Dictionary.com.]