1. Political power

• In the UK
The head of state in the UK is Queen Elizabeth II because it is a monarchy. But as it is a parliamentary monarchy, the Prime Minister (PM) is the head of government and actually holds power, appointing the members of his/her government and governing the country. The Queen only has a symbolic role such as confirming a new PM, she doesn't appoint him/her.

• In the USA
The president of the USA is the head of state because the USA is a republic. The president is also the head of government, which means there is no PM in the USA, because it is a presidential system. However, the president can't appoint the members of his/her government alone: he/she nominates them and the Senate confirms them because of the principle of checks and balances.

2. Political parties

Theresa May, the current UK Prime Minister is a Conservative.
Donald Trump, the current US presiedent is a Republican.

3. Major elections

• In the UK

Holding the election
Since 2011, general elections (GE, legislative elections) are held every 5 year to elect members of Parliament (MPs) to the House of Commons.
=> PM : 5-year term, except in case of a snap election like in May 2017 (instead of May 2020) due to the Brexit negotiations - and to have more MPs...

Becoming PM by winning the general election
In the 2015 UK general election, the Conservative Party won the GE with 51% of the seats in the House of Commons. With more than 326 MPs, the Tories had a simple majority (majorité absolue) but only needed a simple plurality (majorité relative) to win the election.
=> The leader of the winning party, David Cameron, became PM andwith a simple majority in the Commons, he could govern without forming a coalition.

Becoming PM without holding a general election
Conservative PM David Cameron organised the June 2016 Brexit referendum for the UK to stay in or leave the European Union (EU). Cameron supported the Stay option but Brexiteers won the referendum.
=> Consequently David Cameron resigned as PM and party leader. Theresa May was elected leader of the Tory Party and subsequently replaced him as PM without any general election.

• In the USA

Holding the election
The US president is elected every 4 year => 4-year tem. He/She can stand for re-election only once => 2 terms maximum.
Presidential elections are held in November, the winner becomes the president-elect and will officially become president on Inauguration Day in January.
The election is held between the official presidential tickets (president + vice-president -VP) who won their party nominations after a long electoral process starting with party primaries from January to June & ending in the summer party conventions.

Winning the election
In the 2016 US presidential election, Donald Trump, a Republican, was elected president as he won 306 electors, a simple majority of the 538 electors of the electoral college.
What is unusual is that Donald Trump lost the popular vote with 47% of the votes to Hillary Clinton's 48%, but the popular vote is only symbolical and doesn't count because it's an indirect election.