けれども is a polite-ish way to say 'but' or 'although'. It is often shortened in speech to けれど or けど, with けど being the most casual and common in conversation. In writing, けれども is more formal than けど, but still standard.

Use it to contrast two facts or express a mild surprise. For example, you can say something like 'I was tired, but I went out.' The clause with けれども can come either first or second, but in Japanese it usually comes first.

Note that けれども is not as strong as しかし (which is more essay-like). It simply links two sentences softly, and in conversation it's often used just to add information without strong contrast, like saying 'by the way'.