In
phonetics, this refers to sounds made at the same
place of articulation. Literally, with the same
organ, but with most sounds the organ is the
tongue, so in these cases it refers to which point in the
mouth the tongue is touching.
For example, p, b, and m are all homorganic because they are bilabial, made with the two lips. t, d, n, and l are all made at the same place as each other - alveolar in English, dental in French and Italian.
The most common reason homorganic sounds need to be mentioned in phonetics is because languages often assimilate a more distant sound to a nearer sound, because it's easier to say. The articulatory organs don't have to abruptly move from one point to another. So when non-homorganic n and b come together, often the nasal consonant turns into the corresponding homorganic nasal, in this case m. So 'Canberra' may be pronounced as if Camberra.
The strength of this assimilation varies from language to language. Some languages, such as Spanish and Italian, cannot tolerate non-homorganic groups like nb. So San Marco, San Benito, San Carlo are obligatorily pronounced Sam Marco, Sam Benito, Sang Carlo.
In English it is not a strong tendency. We typically assimilate more in casual speech. But we happily tolerate non-homorganic clusters, especially if they belong to different elements: unbroken is unlikely to become umbroken. With some words it varies depending on the individual, e.g. for 'income' some say in-cum, others assimilate it and say ing-cum. Historically, a few words have been affected by it, e.g. ant comes from Old English æmete. Once the mt came together, they underwent assimilation to nt.
Not all kinds of assimilation are of this kind, e.g. the Latin negative prefix in- assimilated to the homorganic sound in im-personal, but assimilated completely to the sound in il-legal, ir-rational.
By far the most common instance of homorganic assimilation is this, where a nasal (n, m) changes to the place of articulation of the following stop. Others are rare. However, in Sanskrit the several kinds of s assimilate to a neighbouring homorganic consonant.