There are many essential items for many tasks, including jogging.

Trainers – ensure your trainers fit properly and are comfortable – it’s worth looking after your feet, as they’ll take the most damage.
Luminous vest – always make sure you are visible, especially at night-time.
Water bottle – ensure you remain hydrated – you can buy bottles in many shapes to make it easier to carry when running.
Summer – shorts, socks – remember, look after your feet! – and running vest / T-shirt
Winter – luminous vest, thermal running top, trackster (much better than jogging bottoms), gloves, socks and windproof jacket.