Replication is the process by which DNA is copied during mitosis and meiosis; transcription is the process by which RNA is synthesized with DNA as a template. These processes have some fundamental similarities, but they differ considerably.
To understand the differences between replication and transcription, it helps to start by understanding the basic similarities. Both replication and transcription occur in the 5 prime to 3 prime direction. Both processes require that the DNA's helical structure be unwound by DNA helicases and kept open by single-strand binding proteins; this is important because the nitrogenous bases that make up DNA's genetic code are naturally inclined toward hydrogen bonding together, and while hydrogen bonds are not normally strong, they become hard to keep apart in the context of DNA's zipper-like structure.
However, behind these fundamental similarities are key differences. The main difference between DNA replication and transcription is scale. DNA replication involves the copying of both strands of an entire chromosome, whereas transcription only involves creating an RNA "mirror" of a small portion of DNA. This is because the purpose of transcription is to put the molecular instructions that DNA gives into action; the RNA that transcription produces is later used by ribosomes to synthesize the proteins that are a cell's building blocks. DNA also requires different polymerases than mRNA for its synthesis. Additionally, in RNA code, the nitrogenous base thymine is replaced by uracil.