In Episode 2, Anakin and Padme go to Tatooine to find Anakin's mother. When they arrive at Watto's place they are in some sort of cart being pulled by a droid.
Does anyone know the name/type of this droid?
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Based on your description, it looks like you are talking about a rickshaw droid:
As seen in Episode II:
Wookieepedia calls it an RIC-series general labor droid, specifically an RIC-920. However, the only reference to this name appears to be from a deleted page on starwars.com (there's an archived version of it here). The novelization of Episode II referred to it as an ES-PSA droid.