Capsicum annuum var. annuum.
These are sweet frying peppers...they are not hot, so don't use as you would chillies.
Shishito peppers are the East Asian variety of the cultivar and are named because the tips of the peppers resemble the head of a lion—shishi (or jishi) in Japanese. On the other hand, Padrón peppers are from Spanish seed of the cultivar from the municipality of Padrón in northwestern Spain, hence its name.
Fridge in a sealed container/bag :)
In the Kitchen
I have only ever had these this way, and honestly, I would not do them any other:
1. Heat a pan with a good drizzle of oil.
2. Throw the peppers in, un-cut, un-anything (leave stems on). Toss to coat in oil.
3. Cook for about 5 minutes, tossing pretty constantly, until they are charred/blackening in parts.
4. Remove from heat, season with salt and pepper, and try to get them from the kitchen to the table without eating them all, I dare you! We eat the stems, but some people prefer to spit them out/tear them off.
Optional: Squeeze of lime, and some sesame oil.