See also the official documentation:
Help Center > Members Manual > Working with group messages

Replying to the Group:

Via Email:

It all depends on how the group is set up. See Message composition tips and tricks for details about types of footers and setup.

If there is a link in the footer to reply to group, you can use that - see Understanding message footers. Depending on group settings, you can reply to the Group or to the Sender (individual member).

Or you can click on Reply in your email program.

Via Group page:
You can reply by going to the group homepage. There is a link in the footer - for example: View/Reply Online (#133). 

A few nice things about replying through the group page:

- drafts autosave. Just look in Messages - it will be obvious.
- it makes it easy to add hashtags.
But you can't see replies you have sent to a moderated group until the message is approved. (true as of Nov 14, 2018). Of course, you can look in Sent mail if you reply through your email program.

Generalities or why didn't my message go through?

Topics can be closed after a certain length of time, so you won't be able to reply.

Some groups are moderated. That means it will take more time before you see your reply appear in the group.

Even in unmoderated groups, the owner / moderator can set a topic to moderate.

Some groups don't allow attachments.

Hashtags:
- Some groups require hashtags. Others don't.
- Some groups are set so that you must use existing hashtags. Otherwise the message is bounced.
- Some groups allow users to create new hashtags.