The Mayans were the first to provide a symbol for the concept of zero. Sometimes it was a hieroglyphic shell, but other times it was a head or other symbol. Mayans have place value in their numbers, unlike the Roman system, but they wrote their numbers vertically, with the "one" digit at the bottom of a column. Their place values used multiples of 20 instead of 10, like the Arabic system. For digits, they used dots, representing one through four and bars for fives. For instance, the number six would have a horizontal bar with a dot resting on top. In the Mayan system, you can write up to the number 19 in this fashion. The number 19 would appear as three horizontal lines with four dots resting on top. The number 86 would have four dots in the 20's place and then, below it, a line with a dot resting on it in the ones place.
Without place value or the concept of zero, the Roman system is the most unique in use today. In the Roman system, different letters are assigned to numbers: 1 (I), 5 (V), 10 (X), 50 (L), 100 (C), 500 (D), and 1,000 (M). A multiplier of 1,000 can be used by writing a bar above any number. The Romans did have a system in place to count as high as they needed. However, those numbers could get really long (1,997 = MCMLXXXXVII) or short (2,001 = MMI). The most common place you can see Roman numerals today is in copyright pages of movies or research outlines.
Using a base-ten numbering system, the Arabic system is very close to the Hindu-Arabic system in common use throughout the world today. Recognizing zero and written from left to right, the only differences between Arabic and Hindu-Arabic numbering are the shapes and names of the numbers themselves. A graphic and audio reading of these numerals can be found on the St. Ignatius High School webpage.
The Roman numbering system was very popular for abacus users, well into the early Renaissance. While the Hindu-Arabic numerals were easier to write out and arrange visually because of place value, they did not win out over the Roman system until the invention of the printing press. The first math textbooks used Hindu-Arabic numerals, and the rest is math history.