Technically, an open reading frame (ORF) can be used to describe sequences in both DNA and mRNA.
A "reading frame," in genomic terms, is a section of a sequenced piece of DNA that is is "marked" for RNA transcription. A reading frame begins with an initiation (methionine ATG) codon and ends with a nonsense codon.
A given stretch of DNA could conceivably be broken up into codons, or chunks of 3 bases, with 3 possible "reading frames." (Think of starting to read the sequence with the first nucleotide, as opposed to the second, or the third, and going by threes.) However, usually only one is the correct reading frame, which will produce the appropriate sequence of amino acids to make up the right protein. The RNA Polymerase enzyme knows where to start transcription because it looks for the sequence "ATG" (this codes for the amino acid methionine and is called a start codon) as a "start signal" for where it should begin reading. Likewise, there will be a "stop codon" at the end of the sequence so the polymerase knows where to stop reading.
When it's time for translation (production of protein from mRNA), the ribosome also uses these "open reading frames" within the mRNA to construct the right sequence of amino acids in a protein. Each codon combination, or set of 3 bases, codes for a different amino acid. A stretch of mRNA, if read in the wrong reading frame, would translate into totally different amino acids, and therefore, into a totally different protein.
Finding a definition of an ORF which refers to DNA is fairly easy, but finding one which describes mRNA is a bit more difficult. So here's a definition of an open reading frame (ORF) for mRNA.
If you need more information, check out this paper I recently wrote on microarrays.