For the longest time, Tumblr had no comments at all, and most Tumblr blogs still do not have any sort of comments enabled.
It is possible to use Tumblr's customizability to insert code from third-party comment systems like Disqus (my personal favorite), IntenseDebate, or LiveFyre to add commenting capability beyond that included with Tumblr's built-in options. If you are really wedded to the idea of using Tumblr, but absolutely want a full-featured comment system, these are the best option.
A hidden benefit?
I want to mention that Tumblr's lack of comments is not strictly a downside, but has some compelling hidden benefits. Tumblr makes reblogging posts easier than on nearly any other major blogging platform, and the combination of the ease of reblogging with the lack of comments means that most of the time, reblogging is used to comment on or respond to posts.
But because reblogging sends traffic, attention, and an implicit endorsement from the commenting blog, this setup tends to mean that people will leave more positive comments than negative, and that the negative comments you receive in this fashion will tend to be limited to people who like and want to support your blog or the overall theme of your post, and just have a small quibble to discuss.
I recommend trying out Tumblr with comments disabled, as I think the lack of comments is actually one of the strong suits of the site; it not only has immediate benefits, but it also shapes the culture of the site in interesting ways.