Translation and explanation of some points in that paragraph:
Homozygous, chicken has two copies of the gene being discussed. That means each chick must inherit one.
Heterozygous, chicken has one copy of the gene being discussed, and one copy of something else. Half of chicks inherit the one gene, half inherit the other gene.
E-locus and Extended Black: the e-locus is a specific spot on the chromosome. The genes at that spot are important for determining how colors are arranged on the chicken. There are at least 5 different genes (each is called an "allele"). One allele is Extended Black, which makes a chicken black all over. Another allele is Birchen, which arranges black & gold on the chicken in a specific way. Yet another is wild-type, sometimes called duckwing, that arranges black & gold in a different specific way. There are also a few others.
Because Extended Black is the most dominant allele at the e-locus, any chicken who has it will look black all over (possibly with a bit of leakage of other colors.) If your rooster is homozygous for Extended Black (has two copies of the gene), then every chick will get one copy of that gene and will look black. If your rooster is heterozygous for Extended Black (has one copy of that gene, and has one copy of a different gene), then half his chicks will be black (inherited Extended Black) and the other half will inherit the other gene (Birchen or duckwing or whatever else he actually has.)
A chicken with Extended Black is black all over unless something changes the black. In this case, barring adds white lines over the black, and Dominant White can turn all the black into white. A chicken who does not have Extended Black will usually have some kind of pattern with black-and-gold, but either the gold or the black or both can be changed to other colors. Gold can be lightened to cream, darkened to red, or completely removed to leave silver (white.) Black can be diluted to blue or chocolate, or changed into white (Dominant White gene).
When Dominant White turns black into white, it sometimes misses bits, so it is called "leaky." That is much more common in chickens that are heterozygous (have one copy of the gene) and less common in chickens that are homozygous (have two copies of the gene.) Some chickens that are heterozygous for Dominant White (only one copy of it) have very obvious black patches, and those chickens are often called "paints." You might get some paint chicks from your white hen and your black barred rooster, if the hen has Dominant White. Or you might get chicks who have little or no black.
When writing about genes, it is common to use abbreviations, such as E for Extended Black. All the alleles (forms of genes) at one locus (place on the chromosome) have abbreviations that start with the same letter. If there are just two, the dominant one gets a capital letter and the recessive one gets a lowercase letter. For example, B is barring, and b+ is not-barred. The + shows that not-barring is the original form of the gene, found in the wild junglefowl. When there are more than two alleles, the abbreviations get an extra letter or two to tell them apart. Sometimes the extra letters are in little raised text, but sometimes they are just marked with ^. So at the e-locus, we have E and E^R and e+ and some others. E is Extended Black, the most dominant of them, and the reason it's called the e-locus instead of the [something else] locus. E^R is Birchen, dominant over most of the others but not dominant over Extended Black. e+ is the wildtype form, sometimes called duckwing or a few other names. (It's got the + to show that it is the original form found in the wild junglefowl, and all the other alleles are mutations that have happened over time.)
If you know what two genes a chicken has, you can write them E/E (chicken has two genes for Extended Black) or E/e+ (chicken has one gene for Extended Black and one for the wild-type form). If you know one gene but not the other, you can write E/? or E/_ (chicken has one gene for Extended Black, but you don't know which other gene it might have.)
Recessive white is recessive, so it only shows when the chicken has two copies of the gene. It turns the entire chicken white. But if you cross one chicken with recessive white (hen) with one that does not have recessive white (rooster), each chick gets just one copy of the recessive white gene, and that is not enough to turn the chick white. That means it shows whatever other genes it has. In this case, the chicks might be black with white barring (from the father). But if he has some other gene recessive to E, then he might give genes that cause some kind of multi-color patterning on the chicks. The hen has two genes at the E-locus too, and we don't know what they are, but they will still be inherited by her chicks-- at which point we might see effects from them.
Let me know if I missed something that still needs translating or explaining
white hen:
