Translation:My little brother is bad at swimming.
I'm still learning, myself, so take what I say with a grain of salt, but:
泳ぎ is the continuative form, also known as the ます-stem. Very often, the ます-stem can be regarded as a noun in its own right; for example, 泳ぎ means "swimming", as in "the activity of swimming". So, you could probably say 私の弟は泳ぎが下手です。 ("My brother is bad at 泳ぎ."). Whether or not it sounds natural to say is another matter.
The の nominalizer follows a verb in one of the plain forms, which includes:
- The plain present/future form, a.k.a. the dictionary form (泳ぐ)
- The plain past tense (泳いだ)
- The plain negative present/future tense (泳がない)
- The plain negative past tense (泳がなかった)
I think there are other forms that could be considered plain forms, as well, although I think they can be regarded as new verbs in one of the above forms (i.e. the potential form 泳げる can be regarded as a new iru/eru-verb and conjugated into one of the four forms above).
kyoudai is usually use to refer someone having a brother or sister, like i have kyoudai, i have brother or sister, and also kyoudai is actully the combination of the kanji oniisan(ani) and otouto, and is read kyoudai, 兄弟、can mean a sibling or the state of being brothers. they also have the sisters, and mostly used to express "they are sisters" and it is the combination of oneesan (ane) and imouto and is read as shimai 姉妹。