It is the alphabet that was used by the Phoenicians. It is at least 3500 years old, and it is assumed that the Greek alphabet was derived from this Phoenician alphabet, as well as the Aramaic/Hebrew, Arabic and a lot of other alphabets.
Like the Hebrew alphabet and Arabic alphabet which were created from it, the Phoenician alphabet had no symbols for vowel sounds. Each symbol represented one consonant. The vowels had to be deduced from context.
As the letters were originally carved into stone, most are square and straight, like characters from the runic alphabet. The writing of Phoenician was from right to left. There are also texts which were written in a boustrophedon manner, 'like an ox ploughs', i.e. from right to left, then from left to right, then from right to left again, etcetera. Various letters have alternative representations, the tav can be written more like a '+' than like an 'x', the chet can have two cross bars, etc.
The Paleo-Hebrew alphabet is very similar to the Phoenician one. Also e.g. Moabitic was written with a similar alphabet. Only after many centuries, the Aramaic alphabet was used for Hebrew, too. Both alphabets consist of only consonants.
The Hebrew names of the letters suggest, that the Phoenician letters were originally pictograms. Some examples: The first letter, aleph, means something like 'ox' or 'cow'. One can still recognize the horns of one in the character. This applies to the capital A in 'our' latin alphabet as well, if one turns it up side down. The word samekh means fish. One can recognize the bones of a fish in the Phoenician character. It is assumed that the Proto-Sinaitic alphabet may have been the predecessor of the Phoenician alphabet.
Phoenician inscriptions were found at various archaeological sites around the Mediterranean where Phoenicians had their colonies, like Biblos (Lebanon) and Carthage (Tunisia).
The chart below shows the alphabet with its nearest equivalents in the Roman al phabet.
The development of the Greek alphabet from the Phoenician alphabet
The Greek alphabet is thought to have developed either directly from the Phoenician alphabet, or to share a common parent in the North Semitic alphabet. The Greeks kept many of the sounds of the symbols, but some which represented sounds in Phoenician which did not exist in Greek were re-used as vowels. Thus the first letter of the Phoenician alphabet, Aleph, bears almost exactly the same name as the first letter of the Greek alphabet, Alpha, but represents a completely different sound.