Let's define the various tools used by BPM Designer to help you track the changes you make to Solution Items.
Check Out: reserve a particular Solution Item for Editing. Any Solution Item which has been Checked Out can only be edited by the person to whom the Item is Checked Out, for any version of the Solution Item. Solution Items that are Checked Out display with an open Lock icon next to them in the Solution Explorer:

Save: commit your configuration changes to the database, as a part of your Check Out. After a Save event, you can safely close BPM Designer and your work will appear in the Solution the next time you open it. However, Save only preserves your changes for you and not for other Users in BPM Designer.
Check In: commit your configuration changes and make them available for other Users. Check In automatically Saves any changes you have made. Solution Items that are Checked In display with a closed Lock icon next to them:

Version Number: a number in three parts to describe the Version of a Solution Item. The Version Number is of the form: Major.Minor.Revision. The Version Number is specific to a Solution Item, so the latest Version of single Solution can contain Process A with Version Number 1.4.7 and Process B with Version Number 2.2.0.
Check In Type: when Checking In a Solution Item, Users can control how the Solution Item's Version is incremented, by choosing the Check In Type: Major, Minor or Revision. The Version Number is then increased by one in the value corresponding to the Check In Type, for each Solution Item you Check In. If you choose a Major Check In Type, the number for Minor and Revision will be set to 0. If you choose a Minor Check In Type, the number for Revision will be set to 0. For example, when you Check In a Process showing a Version Number of 3.4.8 and select the Minor Check In Type, the next Version Number will be 3.5.0.
Undo Check Out: revert the Solution Item to its configuration when last Checked In. Any changes made since the last Check In, even those committed to the database via a Save, will not be preserved in a Version of the Solution Item.
Publish: push the configuration of your entire Solution from the design environment (Workspace) to a particular runtime environment, e.g. Test or Production.
Publish Set: the configuration of your Solution at the time of publishing. Publish Sets are identified by a 3 part number: Major.Minor.Revision, but always increment by a single digit in the Revision.
Solution: a collection of configuration elements, called Solution Items, which support a common purpose, such as Lead Management or Support Tickets. Solutions are managed in Bluespring BPM Designer. The Properties Window of a Solution is considered a Solution Item of the Solution itself, so the Version History of a Solution represents the different Versions of the Solution Properties and not, necessarily, different Versions of every Solution Item within the Solution.
Solution Item: an element of a Solution, such as a Process, Assignment Profile or a System Resource. To Edit a Solution Item, a User must first Check Out the Solution Item. Each Check In of a Solution Item updates the Version Number of the Solution Item.
Runtime Environment: a database and the associated Runtime Services (Notification, Process Engine, Scheduling, Task, et al.) representing a particular functional ecosystem such as Production, Development or User Acceptance Testing.