Put Simply, the role of the Speaker of the House of Commons is to control the House, a place that can become quite rowdy, given the right circumstances.
The Speaker of the House, currently the Right Honourable John Bercow, has several powers at his disposal.
The first is “directing an MP to withdraw remarks if, for example, they use abusive language”. This was most notably seen in recent history when Dennis Skinner called the then Prime Minister David Cameron “Dodgy Dave”. After repeated instruction to withdraw his comment was met with repeated refusal, the Speaker invoked his second power.
“Suspending MPs who are deliberately disobedient - known as naming”, in the case of Mr. Skinner, he was suspended for the remainder of the day’s sitting. See video - https://www.youtube.com/ watch?v=qvIUa47x_Oc
The power most reservedly used by the Speaker, is that of suspending the sitting of the House, and this action is most prevalent when protest erupt within the chamber. This was memorably seen when Tony Blair had purple flour thrown at him from the gallery, and the then Speaker Michael Martin (now Baron Martin of Springburn) suspended the sitting for 1 hour and 12 minutes. However, this was not explicitly identified as an exercise of the powers played out in Standing Order 46, in the Commons Journal. See video - https://www.youtube.com/watch?v=wfuzQrxerZg
Declaring division is another important role of the Speaker. In the House of Commons, verbal votes still take place. The Speaker will begin with, ”The Question is that...", then state the question. Then, the Speaker will say, ”As many as are of that opinion say Aye." After the verbalisation of the “Aye" vote, he calls for the verbalisation of the “No” not by stating, "of the contrary, No”. If the shouts of "No" are clearly more or less than those of “Aye”, the Speaker will then announce their opinion as to the winner - for example, stating, "I think the Ayes have it". Otherwise, the Speaker declares a division by stating “Division, clear the lobby.” See video - https://www.youtube.com/ watch?v=nl5JLohLWI4
The final prominent power, and role of the Speaker of the House of Commons is “asking MPs to be quiet so Members can be heard” - this is often done through the use of the word “order”. The call for “order” will often increase in volume and assertiveness until the House calms. If there is still disorder in the House, the Speaker can address/reprimand the House, and even specific members. See video - https://www.youtube.com/watch?v=8Qu7s-Wox0M
The Speaker of the House of Commons has one of the most important jobs in British democracy, helping to ensure the House is as productive as possible.