After some words like båda (both) and alla (all) you can have either the definite or the indefinite form. For båda, it’s slightly more common with the definite I’d say.
It's the same for singular words? For example = Båda, katten och hunden är mina vänner.
For English speakers, put this link in a separate window side-by-side with the other link provided by stephenbal4 . https://translate.google.com/translate?hl=en&sl=sv&u=https://snackasvenska.wordpress.com/2014/03/11/skillnaden-mellan-bada-och-bade/&prev=search
Both mean 'both' but båda is the pronoun 'both' as in vi båda 'the both of us' or båda är bra 'both are good'. både is the conjunction 'both' and is only ever used together with an och. E.g. både du och jag 'both you and me', hon kan både sjunga och spela 'she can both sing and play'.
Yiddish borrowed it from Germanic languages, primarily German (schmutzig)
And of course it sounds like “smutty” in English, which has taken on a more risqué meaning of dirty :)
Why is "both these shoes are dirty" not an accepted translation given that "skorna" is definite plural?
'Both these shoes' would be båda de här skorna. Båda skorna just means 'both shoes'.