Ogham is a runic script mostly used in Ireland, but to a lesser extent in the northern island of Britain. While the earliest provable use dates from the 4th century AD, there are linguistic clues that it is older: there are two characters H and Z which are in the script but which have not been found in early inscriptions. You might ask how they could tell that the characters were part of the original script if there weren’t examples in the early inscriptions, and the answer lies in Ogham’s regularity.
Ogham is written with a centerline running in the direction of the characters, and strokes crossing the centerline in various patterns. There were originally four patterns: to the left or above, to the right/below, diagonally across, and straight across. For each pattern, there were characters for one, two, three, four, and five strokes. (NB: five other characters were added later.) The “H” was made by putting one stroke to the left or up; the “Z” was made by putting four strokes diagonally across. If the “H” and “Z” were not in the original set, that would upset the tidy symmetry of the alphabet. (“H” and “Z” were seen in later inscriptions.)
There are arguments about where Ogham came from. I prefer the theory that it derived from Latin, since there were Romans running around Britain from 43 AD to 420 AD. The alphabetical ordering is different from almost all other alphabets, but that might be due to the groupings of the characters. The vowels are all in the “straight-across” set of glyphs, and I can imagine that when the Ogham creators put the vowels into their own set, they rearranged everything.
As to why they needed a new script when Latin was there for the taking, the suggestions that I have seen talk about a desire for a script the Romans wouldn’t understand and/or Latin not suiting the language well. However, there are enough bilingual Latin/Ogham inscriptions that it doesn’t seem like they tried very hard to keep it secret, so I have problems with the former. There aren’t any characters in the original set which are not in Latin, so I have problems with the latter. It seems to me that the answer might lie with the technology of the writing system. Latin has lots of curves, which are hard to carve in wood, and harder still to carve on sticks. Ogham is really easy to carve onto sticks, which Ireland had in abundance.
Sticks from 300 AD Ireland unfortunately have not survived. What has survived are carvings on rock tablets. Interestingly, the early writing on stone was not right-to-left or left-to-right, but rather around the edges of the stone. The edge of the stone was used as the centerline of the script.
Interestingly, the names of trees are used for the names of the characters. Legend has it that all the names came from local trees, but some names don’t correspond to any trees known, and some came from trees that are not found in Ireland.