Test Run
Test Run is available on most of the node configuration dialogs at the bottom of the pop-up box. This is to quickly ensure the correctness of a task without running the flow. Test Run applies to both trigger and action nodes.
If the Test Run fails, you must fix the configuration and try again. But once it's successful, the node will mostly run fine during flow execution.

How many records are processed?
Upto 5 records are handled for a Test Run, but this number differs for each node and can also change based on the task configuration. For example, Byteline Web Scraper only scrapes 5 records for a list, but with pagination, 2 pages are scraped with 5 records from each page.
Variable Mapper
The Test Run output is used by the variable mapper to show auto-complete expression for configuring the downstream nodes. So whenever you change the node configuration that can affect the node output data, the Test Run must be run to have the right variables available for configuring other nodes.
Below is a sample variable mapper screen.

Test Run vs. Save & Skip Test
Test Run is mandatory the first time the node configuration is saved. After that, you will get an option to Save & Skip Test. However, if you made a configuration change that can affect the task's data output, you should use Test Run. Otherwise, the node's output variables available for the downstream nodes will not be correct and your flow can run into issues during the actual run.
How to run a complete flow?
After you've successfully executed Test Run for all your flow nodes, you can use "Run Now" to perform a full run for the flow. This option is on the top-right of the flow designer, as shown in the below screenshot.

You also have other options to run a flow, as explained in Running a flow.