I am a physicist by training, however I am now doing computational biology research. I know what genes, DNA, proteins, enzymes, introns and exons are. I sort of understand how DNA is used to create RNA, which is used by ribosomes to manufacture proteins. That is the extent of my biology and genetics knowledge.
Gene isoforms are all the different RNAs that can be synthesised from a single gene. You may be referring to splice variants, which are systems of getting different mRNAs from the same sequence using different combinations of introns and exons.
Translating your wikipedia article: Gene isoforms are the different RNA products obtainable from a gene that can vary on the place where the translation starts (protein synthesis), that have different sequences in the coding region (thus giving different proteins, sometimes with different functions) or by having different untranslated regions, UTR (the overhangs of the sequence that are not translated in the mRNA). This last difference, the UTR, normally gives rise to different stability within the cell as UTR secondary structures usually control the speed of degradation of an RNA.
Hope it helps