Having had a chat with Kurt and RegDwight I will summarize some important thoughts about what comments are for, why we should flag questions, answers and especially comments and I'd like to mention some words about etiquette, although I consider the last point as self-evident.
The post How do comments work? contains all necessary information, but I'd like to highlight some thoughts:
Comments are suitable if a question or answer is incomplete (e.g. if you're missing context), you think the post can be improved but you're unsure, or for giving some advises you consider as important (even if they seem a bit off-topic).
Comments are not for excessive chats and non-relevant stuff, such as Thank you posts without giving any further context. However, nobody will complain if you are a bit chatty or even if you're giving some context relevant jokes (as long as they are not off-color).
If a comment contain meaningful context, it should be integrated into the question or answer, respectively. The comment should be deleted since it is obsolete. Thus, flag the obsolete comment after integration. A moderator will remove the comment. Also flag the comment if you just partly integrated the comment. And flag a comment if you consider this comment as non-relevant (not constructive/off-topic).
Moreover, flag comments that are too chatty. They're just a pain in the neck for every reader.
Read A theory of Moderation for more information about flagging.
Last but not least, some thoughts about etiquette. Sometimes, there are a comments that are a bit pedantic. Even though they stating correct hints; don't forget if you continually leave critical comments they might be considered offensive, since being pedantic. Thus, if you consider being fussy, try to phrase your comment in a positive way.
Though, if an answer is completely wrong (in your opinion), express your thoughts clearly.
If you read a comment which is annoying, don't verbally fight back. This isn't a constructive way. Again, flag the comment.
Please feel free to comment this post if I should integrate some further hints you consider as important. Please create an answer if you want to discuss, disagree or such in a wider context than comments allow.