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Capsicum
The spicy/hot form is also from the genus Capsicum and in Asia you also some nasty spicy ones. I think on Thailand there is the rule: the smaller to hotter ;-)...
The hottest one is from India: Naga Bhut Jolokia with a Scoville rating of up to 1'000'000 (!). Tabasco is just a balmy breeze with only 5000 Scoville...
Sweet Peppers, Bell Peppers or Capsicum is a cultivated species of Capsicum annuum. I do not know if yours is also a Capsicum annuum. There are tons of species around. Origin is Central- and South America, but nowadyas it is cultivated all over the globe.