Rome was a little quiet town on the shores of the Tiber river when her Latin-speaking citizens learned writing from the Etruscans. A few hundred years later, the Romans brought their alphabet to wherever they went (more specifically, conquered). Because of the prestige of Roman culture, many non-Roman "barbarian" nations embraced Latin for court use, and adopted the Latin alphabet to write their own language. Consequently, Western European nations all wrote using the Latin alphabet, and with European imperialism in the last 500 years, the Latin alphabet (with local modifications) is probably the most ubiquitous writing system in the world.
Even though the Latin alphabet is essentially what you're seeing in front of you, the original version was quite different. As Latium (the region where Latin is spoken and Rome is located) and Etruria (the region where Etruscan is spoken) are adjacent to each other, the very first examples of the Latin alphabet resemble the Etruscan alphabet. Nearly all the letters were adopted with the same phonetic values and graphical shapes. Also, the direction of writing was like Etruscan, either right-to-left, boustrophedon, or even left-to-right for about a hundred years during the 6th century BCE (once again influenced by Etruscan fads). On the other hand, the Latins did modify the Etruscan alphabet to suit their language. They threw away the signs , , , , and ([š>], [z], [ph], [th], and [kh] respectively) because Latin didn't have those sounds. On the flip side, Latin also had sounds not present in Etruscan. One solution was to invent the letter G by adding a vertical stroke to the letter C. Similarly, the Latins "resurrected" the letters O and D, which were not used in Etruscan but kept for tradition. The letter F, which in Etruscan represented the sound [v], was eventually reused for [f]. Etruscan wrote the [f] sound with the digraph HF, a convention also used in the earliest Latin inscriptions. The Latins also took Q and used it for their [kw] sound most likely since it already appears in front of V in Etruscan.
Slowly the Latin alphabet became increasingly standardized. Writing direction settled on left-to-right toward the 5th or 4th century BCE, and letter shapes became more or less the same in Latium. And by Rome's Republican period (3rd century BCE), the Latin alphabet has evolved to the "modern" form:
The letters Y and Z were added for to write Greeks loan words during the early Imperial period (1st century BCE). With these two additions, the Latin alphabet at the late antiquity was nearly identical to most Western European alphabets. During the middle ages, new letters were created by slightly modifying an existing letter. The letter I was used for both [i] and [y], and so J was created from to represent the [y] semi-vowel. Similarly, V doubles as [u] and [w], and so U was created to denote [u] while V stood for [v] only. Still later, the letter W was created in Germanic-speaking regions from doubling U to represent the [v] sound (while the letter V shifted to represent [f]). This doubling trick is also found in other places such as Spanish where the letter Ñ originated from the NN.
As you might have noticed, the classical Roman Latin alphabet only has what we called "upper case", or majuscule, letters. So where did "lower case", or minuscule, letters come from? By the 4th century CE, a semi-cursive style called uncial was being used for handwriting. Uncial is considered a majuscule style but with rounded letters. Eventually this evolved into the minuscule style by the 8th century CE. Originally the two styles were used separately, majuscules for monumental inscription, and minuscules for manuscripts. However, during the reign of Charles the Great (early 9th century CE) the Carolingian Reform forced the merging of the two styles and the creation of the "dual alphabet". With this, our modern Roman alphabet was born.