Why is it called a series?
See Earliest Known Uses of Some of the Words of Mathematics (Series):
SERIES. According to Smith (vol. 2, page 481), "The early writers often used proportio to designate a series, and this usage is found as late as the 18th century."
According to Smith (vol. 2, page 497), "The change to the name ’series' seems to have been due to writers of the 17th century. Even as late as the 1693 edition of his Algebra, however, Wallis used the expression 'infinite progression' for infinite series."
In the English translation of Wallis' Algebra (translated by him and published in 1685), Wallis wrote:
Now (to return where we left off:) Those Approximations (in the Arithmetick of Infinites) above mentioned, (for the Circle or Ellipse, and the Hyperbola;) have given occasion to others (as is before intimated,) to make further inquiry into that subject; and seek out other the like Approximations, (or continual approaches) in other cases. Which are now wont to be called by the name of Infinite Series, or Converging Series, or other names of like import.
Series comes from Latin series, from serere (“to join together, bind”).
To mention more specifically what some of the comments above are referring to: series do not have many of the properties of sums, so the name "infinite sum" would be misleading. A notorious example of this is that in a series, the value of the series can depend on the order in which we "sum" up the terms. There is a famous theorem called the Riemann rearrangement Theorem which says that if a series is conditionally convergent (i.e. the series converges but it does not converge absolutely) then one can permute the order of the terms of the series to make the series add up to any value you like. See for example http://individual.utoronto.ca/jordanbell/notes/summable.pdf for a proof of this.
A series is a sum that results from adding up terms of a sequence (which has a well defined order). One should understand that mathematical concepts were viewed differently in the past - the notion of infinity, which we now take for granted, was not thrown around lightly, and notation also wasn't what it is now.
The same root as series (which is explained by @Mauro to come from Latin) is shared by all major romance languages I checked. In German, other Germanic languages, and most Slavic languages, the equivalent of the word order is used (and matches the word for order used for taxonomic rank in biology). In Slovenian, the word is vrsta, which loosely translates as queue/line-up (or possibly sort/species, but in this case the similarity with taxonimical classification is a coincidence).