Updated 10th June:
The General Election held on 8th June 2017 produced what is referred to as a "Hung Parliament" - that is, one in which there is no political party in the House of Commons with an outright majority over all other parties. The outcome was that the Conservative and Unionist Party won the most seats in the Commons. The result may be seen at BBC - Election 2017. The 5 parties with the most seats were:
Conservatives - 318 seats (loss of 12) - 42.4% Vote share
Labour - 261 seats (gain of 29) - 40.0% vote share
Scottish National Party (SNP) - 35 seats (loss of 21) - 3% vote share
Liberal Democrats - 12 seats - 7.4% vote share
Democratic Unionist Party (DUP) - 10 seats - 0.9% vote share. (A party from Northern Ireland).
In the Parliament of 2015-17 the Conservatives had a majority. The June 2017 election was held because the Prime Minister (Rt Hon Theresa May MP) wished to increase the majority held by her party so as to strengthen her hand in the forthcoming Brexit negotiations. The Fixed-term Parliaments Act 2011 was followed - (earlier post) - and the election proceeded. The outcome was loss of the majority held by the Conservatives. The Labour Party - led by Rt Hon Jeremy Corbyn MP - undoubtedly fared better in the election than most expected and became the second largest party in the House of Commons which allows Jeremy Corbyn to continue as Leader of Her Majesty's Opposition.
One immediate political problem with a "hung parliament" is to find a government able to present a programme to Parliament (the Queen's Speech) and to have it accepted (Debate on the Queen's Speech). It is possible for the Queen's Speech to be rejected by the House of Commons but this has not happened since 1924. Those events, involving the government of Stanley Baldwin, are described here. The House of Lords also debates the Queen's Speech but does not vote on it.
As leader of the largest party in the Commons, Theresa May has the opportunity to try to form a government and she can seek co-operation from other parties so that a programme for government can be presented in the Queen's Speech. At the time of writing (9 - 10th June), she might have been successful in doing this because a Northern Ireland party - the Democratic Unionist Party (DUP) - appear to be willing to support a programme though probably not as a coalition (see here). It would be a "confidence and supply" arrangement and the details of what they would support will have to be hammered out. The political "price" for such any such support remains to be seen.
On the morning of 9th June, Theresa May went to see HM The Queen - in law the government is Her Majesty's Government - and then returned to 10 Downing Street to get on with the task of forming the new government. Read the Prime Minister's speech following her return to Downing Street.
If the DUP enters into an arrangement with the Conservatives, the DUP stance on many issues is likely to become significant and may be thought to be out of proportion to the 10 seats it holds. Brexit negotiations with the EU are due to commence and it may be that the DUP will bring about a different approach to the negotiations - (see below where the DUP manifesto is considered).
Readers - especially those outside of the UK - may wonder about the House of Lords in all of this. The Lords is an entirely unelected body and so the election as such has not altered its composition. There are likely to be "Dissolution Honours" and these may include the creation of further peers who will obtain the right to sit in the Lords. Often such peers have been notable members of the House of Commons.
The Cabinet Manual (and here) contains information about the formation of a government in the event of "hung parliaments." The Manual does not have the force of law.
It is fair to say that, outside of Northern Ireland, the DUP as a party and its politics are not well known - see Information about the Democratic Unionist Party . Their 2017 Manifesto covers Restoring Devolution (the Northern Ireland Assembly is not functioning at the time of writing), Increasing family incomes, Creating a Globally Effective Economy, Improving Public Services, Defending our nation from old and new threats, Getting the best deal for Northern Ireland from the UK leaving the EU and a Real Respect Agenda.
The section on Brexit contains 30 priorities and objectives which the DUP wishes to see as part of Brexit negotiations. There can be little doubt that the DUP will use its influence on the UK government to press for those.
See also Election 2017 - DUP
Restoration of Power-sharing in Northern Ireland:
The Northern Ireland Assembly was dissolved in January 2017 and elections were held on 2nd March 2017. There are 90 seats in the Assembly and the DUP won 28 of them making it the largest party in the Assembly - Election results. Since then, power-sharing, a key feature of the Good Friday (Belfast) Agreement, has not resumed and a date of 29th June has been set - BBC News 29th April - for its restoration. The UK Parliament passed legislation to extend the timescale for the making of Ministerial appointments in Northern Ireland - Northern Ireland (Ministerial Appointments and Regional Rates) Act 2017. The Explanatory Notes to the Act offer a good summary of the reasons for the legislation.
An interesting feature of the election is the reduced influence of the Scottish National Party (SNP) though the Party continues to hold power in the Scottish Parliament and Government. The loss of SNP seats was mainly to the benefit of the Conservative Party. This appears to push the prospect of a further Scottish Independence Referendum well into the future though it is not possible to be certain about such matters.
As for the Labour Party - they will be the largest part of the Opposition in the House of Commons but they are unlikely to get the opportunity to form a government unless it transpires that the Conservatives (even with DUP support) are unable to get the Queen's Speech accepted. Let's leave that for now and wait and see what happens! The Labour Party is reported to be preparing an alternative Queens Speech - Independent 10th June.