How can I determine the status of a job?

You could try using the system stored procedure sp_help_job. This returns information on the job, its steps, schedules and servers. For example

EXEC msdb.dbo.sp_help_job @Job_name = 'Your Job Name'

SQL Books Online should contain lots of information about the records it returns.

For returning information on multiple jobs, you could try querying the following system tables which hold the various bits of information on the job

• msdb.dbo.SysJobs
• msdb.dbo.SysJobSteps
• msdb.dbo.SysJobSchedules
• msdb.dbo.SysJobServers
• msdb.dbo.SysJobHistory

Their names are fairly self-explanatory (apart from SysJobServers which hold information on when the job last run and the outcome).

Again, information on the fields can be found at MSDN. For example, check out the page for SysJobs

This is what I'm using to get the running jobs (principally so I can kill the ones which have probably hung):

SELECT
    job.Name, job.job_ID
    ,job.Originating_Server
    ,activity.run_requested_Date
    ,datediff(minute, activity.run_requested_Date, getdate()) AS Elapsed
FROM
    msdb.dbo.sysjobs_view job 
        INNER JOIN msdb.dbo.sysjobactivity activity
        ON (job.job_id = activity.job_id)
WHERE
    run_Requested_date is not null 
    AND stop_execution_date is null
    AND job.name like 'Your Job Prefix%'

As Tim said, the MSDN / BOL documentation is reasonably good on the contents of the sysjobsX tables. Just remember they are tables in MSDB.

-- Microsoft SQL Server 2008 Standard Edition:
IF EXISTS(SELECT 1 
          FROM msdb.dbo.sysjobs J 
          JOIN msdb.dbo.sysjobactivity A 
              ON A.job_id=J.job_id 
          WHERE J.name=N'Your Job Name' 
          AND A.run_requested_date IS NOT NULL 
          AND A.stop_execution_date IS NULL
         )
    PRINT 'The job is running!'
ELSE
    PRINT 'The job is not running.'

I used the top-rated answer to create a simple SQL Function to check if a SQL Agent Job is already running:

-- ===================================================================================
-- Function: "IsJobAlreadyRunning"  |  Author: Geoff Griswald  |  Created: 2021-05-06
-- Description: Check if a SQL Agent Job is already Running - Return 1 if Yes, 0 if No
-- ===================================================================================
CREATE FUNCTION dbo.IsJobAlreadyRunning (@AgentJobName varchar(140))
RETURNS bit
AS
BEGIN
DECLARE @Result bit = 0
     IF EXISTS (SELECT job.name
                  FROM msdb.dbo.sysjobs_view job
            INNER JOIN msdb.dbo.sysjobactivity activity ON job.job_id = activity.job_id
            INNER JOIN msdb.dbo.syssessions sess ON sess.session_id = activity.session_id
            INNER JOIN (SELECT MAX(agent_start_date) AS max_agent_start_date
                          FROM msdb.dbo.syssessions) sess_max ON sess.agent_start_date = sess_max.max_agent_start_date
                 WHERE run_requested_date IS NOT NULL 
                   AND stop_execution_date IS NULL
                   AND job.name = @AgentJobName)
   SET @Result = 1
RETURN @Result
END;

This is useful because I can call this function and check if the job I want to start is already running before I attempt to start it, without adding a lot of bloat to my code. for example:

DECLARE @JobName varchar(140) = 'MyAgentJobName'
     IF (SELECT dbo.IsJobAlreadyRunning(@JobName)) = 0
   EXEC msdb.dbo.sp_start_job @JobName

You haven't specified how would you like to see these details.

For the first sight I would suggest to check Server Management Studio.

You can see the jobs and current statuses in the SQL Server Agent part, under Jobs. If you pick a job, the Property page shows a link to the Job History, where you can see the start and end time, if there any errors, which step caused the error, and so on.

You can specify alerts and notifications to email you or to page you when the job finished successfully or failed.

There is a Job Activity Monitor, but actually I never used it. You can have a try.

If you want to check it via T-SQL, then I don't know how you can do that.

Below script gets job status for every job on the server. It also tells how many steps are there and what is the currently running step and elasped time.

SELECT sj.Name,
    CASE
        WHEN sja.start_execution_date IS NULL THEN 'Never ran'
        WHEN sja.start_execution_date IS NOT NULL AND sja.stop_execution_date IS NULL THEN 'Running'
        WHEN sja.start_execution_date IS NOT NULL AND sja.stop_execution_date IS NOT NULL THEN 'Not running'
    END AS 'RunStatus',
    CASE WHEN sja.start_execution_date IS NOT NULL AND sja.stop_execution_date IS NULL then js.StepCount else null end As TotalNumberOfSteps,
    CASE WHEN sja.start_execution_date IS NOT NULL AND sja.stop_execution_date IS NULL then ISNULL(sja.last_executed_step_id+1,js.StepCount) else null end as currentlyExecutingStep,
    CASE WHEN sja.start_execution_date IS NOT NULL AND sja.stop_execution_date IS NULL then datediff(minute, sja.run_requested_date, getdate()) ELSE NULL end as ElapsedTime
FROM msdb.dbo.sysjobs sj
JOIN msdb.dbo.sysjobactivity sja
ON sj.job_id = sja.job_id
CROSS APPLY (SELECT COUNT(*) FROM msdb.dbo.sysjobsteps as js WHERE js.job_id = sj.job_id) as js(StepCount)
WHERE session_id = (
    SELECT MAX(session_id) FROM msdb.dbo.sysjobactivity)
ORDER BY RunStatus desc