All of these factors are linked through what is called the central dogma: DNA is transcribed into mRNA, and mRNA is translated into a polypeptide, the precursor to a protein.
So, we know that we start from DNA, and have to get that DNA "out" in some way, so that it is expressed in the "final product". We might presume the product is a single polypeptide, or go so far as to refer to the entire organism.
We'll start with the gene. A gene is a "unit of heredity"; basically a small chunk of DNA that codes for one particular thing. Genes can be composed of just a few nucelotides, or many; the average human gene has somewhere around 100 nucleotide pairs. If your DNA is like a cookbook, a gene is one page, or "recipe" from that cookbook.
The word "genome" has two uses. It usually refers to all of the DNA in your body (the cookbook) but it can also be used to refer to ALL of the variations that an entire species possesses. However, to make things simple, we'll just refer to a single organism. So, we can see that DNA is organized into genes, each of which codes for a single trait, and the genome is all of the genes in your DNA.
In order to actually do anything with all those genes, we need to get their information out of the protective coating of the nucleus and transmit it somehow. We do this through messenger RNA, or mRNA. In eukaryotes (organisms with a complex, nucleated cell, like us), there's an extra step in this process; the pre-mRNA.
mRNA is really just a copy of a gene; it's just like you wrote down the instructions from the cookbook. However, the process is a bit more complicated than that; in order to make a copy, RNA needs to bond to the nucleotides inside the gene. The gene accomplishes this by having the DNA "unzip" itself, just like a zipper, splitting the bonds between the nucleotides and allowing a process called transcription (or copying) to take place. The RNA has a lot of mobility, and it can leave the nucleus and travel to other parts of the cell. This is also an excellent defensive technique for the cell; by keeping the original copy of the DNA inside the protective nucleus, the cell lowers the chance that the DNA will be damaged or corrupted.
Pre-mRNA and mRNA differ in a suprising way. Pre-mRNA is an exact copy of the gene, nucleotide by nucleotide. However, it undergoes a process called splicing and capping. This involves removing chunks of the pre-mRNA that are not actually used to produce a polypeptide. These chunks are called introns, and the useful chunks are called exons. The "mature" mRNA contains only exons; this is what actually determines the polypeptide's composition.
So, to complete our cooking analogy; the genome is like a cookbook full of recipes. One recipe is a gene. If I copy that recipe word for word, i've made pre-mRNA. If I simplify my copy down to the bare essentials, that's mRNA.
Source: Biology, Second Edition. Brooker, Widmaier, Graham & Stiling, 2008