The size of the buy-ins your bankroll allows depends on several factors. Among these the variant you play, if it is PLO, your bankroll must be higher than that required if you play No Limit Hold'em because in Omaha the variance is greater. But it also depends on your playing style, a wide aggressive game requires a higher money pool than a tight-aggressive style.
The bankroll required will also vary depending on whether you play in a tournament, cash game or sit n go. It is commonly accepted that MTT players need a larger bankroll as the swings can be significant.
The table below recommends the number of buy-ins required for the majority of games available on the internet (note that the bankrolls required live and online are not the same).
Regarding tournaments, the figures mentioned below are the number of buy-ins. So if you play €5 MTTs, a minimum of 100 buy-ins is required, i.e. a bankroll of €500. For cash games, this is the maximum buy-in limit: if you play at a six-player NLHE table, €0.10/0.25, whose maximum buy-in is €25, it is preferable to have €2,500.
gambling (online) Minimum Medium Cautious
No-limit hold'em (6-max.), cash game 30 buy-ins 50 buy-ins 100 buy-ins
Hold'em without limit (full ring), cash game 25 40 75
Omaha Pot-Limit (6-max.), cash game 50 100 150
Omaha Pot-Limit (full ring), cash game 30 50 100
No-limit hold'em, 9-player sit-n-gos 30 50 100
No-limit hold'em, 45-player sit-n-gos 50 100 150
No-limit hold'em, 180-player sit-n-gos 100 200 500
No limit hold'em, multi-table tournaments 100 200 500
No-Limit Hold'em, multi-table tournaments (large field) 200 400 600