The short answer to your question is this: there is no analog to
apex:pageBlock or its child tags in Lightning Components.
As you know, the
pageBlock family of tags in Visualforce gave developers tools to create Salesforce look-and-feel pages. There were a bunch of attributes and options to make it so that you could compose a UI that looked just like a page layout, but without actually conforming to page layout metadata. No understanding of CSS, or the underlying actual HTML composition was required.
This has some very positives: once you understand how it works, and if you are willing to play by the rules of how Salesforce composes HTML for you, you can very quickly put together a standard SF-looking UI.
But it has some down sides: if you want to break the rules the slightest bit, there is potentially a lot of work to do. In fact, a lot of developers have devoted a lot of time to reverse-engineering the Salesforce UI when they wanted to do this.
In the world of Lightning Components there are, as of Winter 16, a number of features in VF that have not yet shown up. Bringing in a
pageBlock analog is obviously one of them. But actually Salesforce have taken a different approach as a first iteration of recreating the standard UI
Instead of making a set of components in LCF that will just build the UI, the actual CSS and UI framework are themselves a supported feature.
Salesforce Lightning Design System (SLDS) is the Salesforce look and feel. If you want to build a UI that looks like Salesforce, there is how. There are lots of examples that show you how to go about it. It isn't complete, but the UX team is adding new examples all the time, and new versions are being released pretty regularly.
But, the downside here is that this only applies to the new Lightning Experience (LEX) desktop functionality (along with a few mobile examples). It will not look like the pre LEX look and feel, or, what is sometimes referred to as the "Aloha" UI. Which is what
apex:pageBlock used to give you.
For specifically getting SLDS working with Lightning Components, there are examples already in the SLDS web page, and another answer to this question to show one way to do this.
My hope is eventually they give us some kind of component in LCF that will just show either the Aloha, or LEX UI, based on user config. But I suspect, if that ever happens, that will be a long way off.