Depending on the context, the Spanish phrase 'mucho gusto' can have several different meanings when translated in English. The phrase directly translates to 'much pleasure', and it's common to see the phrase 'con mucho gusto', which means 'with much pleasure'. Contexts where 'mucho gusto' appears include introductions and expressing thanks.
'Mucho gusto' is commonly used during polite introductions, to express gladness that you have met someone. An English equivalent would be the phrase 'pleased to meet you'. For example, say a person named Lesley is meeting a new coworker, Nick for the first time. Their introductions could go something like this: Lesley: Hola, me llamo Lesley. (Hello, my name is Lesley.) Nick: Mucho gusto, Lesley. Me llamo Nick. (Pleased to meet you, Lesley. My name is Nick.) The related phrase 'con mucho gusto' is used in response to a person expressing gratitude. An equivalent phrase in English is 'with pleasure'. For example, if Nick expressed thanks for a gift Lesley gave to him, Lesley could respond 'con mucho gusto', which would give the same sort of sense as the English sentence 'It was my pleasure'.