I recently learned that the UK House of Lords has 26 bishops as well as a number of hereditary members. Do these Lords of Parliament have real political power or are they largely symbolic like the Queen?
All those who have the right to sit and vote in the House of Lords have the same rights and privileges, and yes, they do use them.
To take an example which is topical at the time of writing, this division (vote) on an amendment to the Brexit bill included both bishops and hereditary peers.
The bishops are not associated with any political party, and we can see in this case that 2 voted for the amendment, and 8 against - including the Archbishop of Canterbury, who is the most senior bishop in the Church of England.
A number of the 92 hereditary peers who remain in the House voted, including (if I've counted correctly) 2 Dukes, 1 Marquess, 21 Earls/Countesses, and 18 Viscounts. Some hereditary barons and baronesses may also have voted.
A quick scan of Hansard (the official transcript) from an earlier stage of the bill shows that the list of peers who spoke in that debate also included both bishops and hereditary peers.
While the House of Lords is not wholly impotent, its collective power is largely limited to delay (by up to a year) and to making suggestions to revised bills in a manner similar to that of a U.S. legislative committee staff. It can also act as a "bully pulpit" for notable aristocrats who wish to make political statements. It can in theory introduce legislation, but that legislation goes nowhere if the Prime Minister and his cabinet don't support the legislation.
Historically, it has a judicial role, carried out by law lords, but that was largely discontinued when the Supreme Court of the United Kingdom was created in 2009. In particular, the Privy Council of the United Kingdom was the court of last resort for all members of the British Commonwealth.
The House of Lords also continues to have a meaningful role in the internal affairs of the Church of England (which once had significant temporal powers over probate law, family law, welfare law, education provision and certain aspects of criminal justice), and in regulating the royal family in terms of family law and succession (e.g. royal marriages and divorces can require parliamentary assent, IIRC).
The House of Lords scrutinises bills that have been approved by the House of Commons. It regularly reviews and amends Bills from the Commons. While it is unable to prevent Bills passing into law, except in certain limited circumstances, it can delay Bills and force the Commons to reconsider their decisions. In this capacity, the House of Lords acts as a check on the House of Commons that is independent from the electoral process. Bills can be introduced into either the House of Lords or the House of Commons. Members of the Lords may also take on roles as government ministers. The House of Lords has its own support services, separate from the Commons, including the House of Lords Library.
The Queen's Speech is delivered in the House of Lords during the State Opening of Parliament. In addition to its role as the upper house, until the establishment of the Supreme Court in 2009, the House of Lords, through the Law Lords, acted as the final court of appeal in the British judicial system. The House also has a Church of England role, in that Church Measures must be tabled within the House by the Lords Spiritual.
What are Church Measures?
The Church of England Assembly (Powers) Act 1919 (9 & 10 Geo. 5 c. 76) is an Act of the Parliament of the United Kingdom that enables the Church of England to submit primary legislation called Measures, for passage by Parliament. Measures have the same force and effect as Acts of Parliament. The power to pass measures was originally granted to the Church Assembly, which was replaced by the General Synod of the Church of England in 1970.
The 92 hereditary Lords who until reforms in 1999 predominated over Life Lords have the same rights as Life Lords, and are at no disadvantage relative to Life Lords who are, if anything, inferior in status to Hereditary Lords.
In general, all members of the House of Lords have equal ability to vote and participate in its deliberations, although a minority of a couple hundred do so in practice. Church measures pertain mostly to the internal governance of the Church of England and so far as I know continue to be passed on a relatively routine basis over those matters, generally initiated by one of the Lords Spiritual.
The House of Lords is certainly not fully subordinate to the Prime Minister and House of Commons in the same manner as the Queen. Subject to its formal limitations on power, it does act independently, mostly either on technical matters to avoid bad legislative drafting and when issues of great national importance such a human rights and sovereignty come up where its bully pulpit powers can have a meaningful effect on the House of Commons.