Recently, when FC Barcelona signed Arthur from Brazil, reports regarding another new signing Yerry Mina being sent on loan due to some foreign player restrictions surfaced. What is this rule and do similar rules exist in Europe's other top leagues?
La Liga regulations permit up to three "foreign" players per team.
"Foreign" refers to a player from outside the European Union. Due to a legal ruling over a handball player in Germany, there are exceptions for people from countries with EU association agreements (because a quota that counts them is a restriction on their employment opportunity contrary to such an agreement).
Players with ancestry of an EU country may also use this to be counted as an EU player, and therefore not as foreign. For example, Argentine players of Italian descent have used this method to avoid the quota.
Germany's Bundesliga 1 and 2 do not have a maximum foreign player limit, but instead a minimum "local" player limit. At least 8 players must have played for the club in three seasons each before turning 25, and at least 4 further players must have done the same at any German club.
France's Ligue 1 appears to have a limit of five foreign players (with a similar meaning as above) and further allows only four to be playing on the field at any time. However sources on this are unclear as to whether the five is actually four (and thus the second limit is unnecessary). Actual regulations are not easy to find.
English Premier League currently requires a minimum of 6 "home-grown" players in the match list of 18, which is expected to rise to 7 and additionally require a "club-grown" player in 2018-19.
Notably this rule explicitly disregards nationality or citizenship. Such players must have played three seasons or three years for an English or Welsh-affiliated club before or during the year they turn 21.
Currently, 3 spots are reserved for Philippe Coutinho, Paulinho and Yerry Mina (Other originally non-EU players, namely Messi worked more than 5 years in Spain and acquired Spanish citizenship; Suarez has Italian citizenship through his wife). Now, if Arthur joins the club, someone has to either leave or gain European citizenship. There are rumors that Coutinho will apply for Portuguese citizenship through his wife to free up one space.
The foreign player quotas of other top European leagues are well documented in @Nij's answer. These discriminating quota rules are actually not just a football thing and are practiced in other sports as well. Here you can find a paper by M. Greenberg and J. Gray "Citizenship Based Quota Systems in Athletics" which describes some citizenship based quota systems in sports and their expected effects on competitiveness.