You need . . . . . . . . visa to visit some countries, but not all of them.

The article 'the' is used when the noun is specific and known to the speaker as well as the listener. The noun 'visa' is not specified as to which visa and it's been mentioned for the first time here, so 'the' isn't the correct article. An indefinite article, either 'a' or 'an' is required. 'An' is used for nouns beginning with a vowel, 'visa' begins with a consonant, and hence 'a' is the correct article for 'visa'.