Certain consonants change when you add an ending that puts a vowel after them (like the accusative ending here). K is one of them, and it becomes ğ. (Also, p becomes b, t becomes d, and ç becomes c.)
There's a section about it in the Accusative tips and notes: https://www.duolingo.com/skill/tr/Accusative (scroll down to "Consonant Mutations" at the bottom).
They mean different things. "You eat chicken" uses chicken as a general direct object. You use it when you're speaking generally -- for instance, if you're saying "you eat chicken" to mean that chicken is a food you eat. But in Turkish, a general direct object doesn't take an accusative ending, so you'd use "you eat chicken" as a translation of "siz tavuk yersiniz."
The Turkish sentence here, "siz tavuğu yersiniz," does have an accusative ending, so that indicates that they're using chicken as a specific noun. In English, that's "you eat the chicken," and it refers to particular chicken (maybe you eat the chicken they serve at that restaurant, or the chicken that's in our refrigerator).