DNA stands for deoxyribonucleic acid. It is composed of three major parts: the central deoxyribose sugar, a nitrogenous base, and a phosphate group. The numbering system of 3' and 5' corresponds to the numbering system on the ring of the ribose sugar. The 5' end contains a phosphate group and the 3' end contains a hydroxyl group. Since double helical DNA is composed of two complimentary strands, one of the strands is oriented 5' to 3' and the other strand is oriented in the opposite fashion (3' to 5').
Now let's talk a bit about DNA replication. When strand of DNA is being replicated, the opposing nitrogenous bases that are normally hydrogen bonded together are pulled apart into the two individual sequences of DNA. As the strand is being pulled apart, it forms a "Y" type shape. You can think of it as a zipper being slowly unzipped. The branching point of the "Y" is called the replication fork. As more of the DNA is "unzipped," the replication fork moves slowly down the strand. The leading strand of DNA is the sequence of DNA that is being replicated in the same direction as the fork. Since replication always occurs in the 5' to 3' direction, that means that the leading strand ends with the 3' end since DNA binding is complementary. The lagging strand is the sequence of DNA that is being replicated in the opposite direction as the replication fork. It ends with the 5' end, the opposite of the leading strand. A diagram of this is shown in the links below.