In the second part of the تشهد tashahhud, we continue to ask Allah to bless our Prophet (peace be upon him) as an acknowledgement of our gratitude to him for conveying to us the message of our Lord, and so we say, اللهم صل على محمد وعلى آل محمد usually translated as, “O Allah, send prayers upon Muhammad and on the family of Muhammad.”
The verb form ِّصل is a way of requesting someone to ‘pray’, but when we ask this of Allah we are requesting that He sends His special mercy and blessings upon the Prophet (peace be upon him) through honouring him both in his world by elevating his remembrance and manifesting his call, and in the Hereafter by interceding for his nation, and multiplying his reward.
When we refer to آل محمد - we are referring to not just to his pious and noble family who believed in him as a Prophet and shared the hardships of life along with him, but also all those who are his family in the faith - his companions and followers who sincerely obeyed his command and tried to believe by the ideals prescribed by him, until the Day of Resurrection. So in this portion of the tashahhud we are effectively praying for ourselves.
So when we say, اللهم صل على محمد وعلى آل محمد we are essentially saying, “O Allah, send Your special mercy and prayers upon the Prophet (peace be upon him) in the most complete, encompassing way, both in this world and in the Hereafter, and also send Your mercy and prayers upon all those who are affiliated to him, be they from his family, his companions, or his followers in Islam.”