The flesh of mangoes differs from other fruits typically used for juicing. Unlike in citrus fruits, the membranes that hold mangoes' juice, or vesicles, aren't plump, so juice pressers and centrifugal juicers yield little, if any, juice. You have to shred mango flesh into small pieces, or comminute it, to release the juice in its vesicles. Unless you have a masticating juicer, the type that grinds instead of spins, use a basic blender or food processor to juice a mango, then strain the pulp out in a separate step.
Peel the mango and place it on a paper towel to keep it from slipping. Cut the flesh from around the seed and chop it into 1-inch pieces.
Add the mango to a blender or food processor,along with about 1/2 teaspoon of lemon juice to prevent oxidation. Add 1 cup of water to the blender and, if you want chilled juice, a few ice cubes. If you want to sweeten the mango, do so now; start with 1 tablespoon of sugar, honey or agave per mango and add more later, if necessary.
Puree the mango on high until smooth. Pour the pureed mango through a sieve or mesh strainer lined with 2 layers of cheesecloth and into a bowl. Use a wooden spoon or rubber spatula to press the juice through the sieve.
Taste the juice. Dilute it, if necessary, by whisking in cold water until it reaches the desired consistency.