The Lightning Page Layout can be customized by your system administrator in a number of ways (by adding or removing various lightning components),  But here is what you might generally see:

1. Highlights Panel -- an at a glance view of the most important fields in the record.

2. Related List Quick Links -- all the related objects that have a relationship to this record are listed, as well as the number of related records.  So for example, we an see this contact, Art Trout has 10 Connection records.  Hovering over a quick link will expand it showing some of those records.  You can then click on any of the related records and go directly to that record.  The drop down triangle to the right gives you the option to edit the related record.

3.  The main page layout for the record.  This is made up of sections, and fields.  Each field can be edited and updated (unless its a read-only field).  Fields with a pencil icon can be double clicked on and edited inline.  Or if you want to edit the entire record (multiple fields), click on the Edit button (4).

5. Lightning components.  In this case, we have a 'page' component which has two tabs.   Thew Activity Tab allows you to email this record, log a call, or create a task. The Chatter Tab allows you to use Salesforce's chatter feature to share information about the record socially throughout your organization.  (You can subscribe to chatter feeds!)

See this Trailhead module for information on using Salesforce Chatter in Lightning.

A record detail page has two sections:

Record Header - is where the fields for this record are laid out.   It is available in both DETAIL view and EDIT view.  Inline editing is available for these fields.

Related Lists - are detail view only with hyperlinks.  You must go to a related object before you can edit it.  There is no inline editing.

1. Chatter about this record: - You can choose to hide this.

2. Related Lists - (with hoverlinks)

3. HandsOn Connect Dashboard -- custom 'at a glance' information associated with many HandsOn Connect objects

4. Section - the record header is broken into individual sections, to group fields in a useful way.  Each field has a label identifying it, and then the data itself.  Double clicking on a field name will allow you to edit it inline.

5. Related Lists - a handy view of objects that are related to this record, with hyperlinks to go to those records.

In EDIT mode - you have access to edit all the fields at once

By double clicking on an individual field, the pencil icon will appear, and you can can edit one field at a time.  You can click the looping arrow to UNDO.

Fields that are locked against editing will have a padlock next to them.

Changes made by inline editing appear in orange.  They are not saved until you hit the SAVE button to save the record.