This is a great question! I'm gonna try and break it up into chunks connected with your subquestions.

First off, your question concerns the British royalty, not nobility which is a separate class but they are very much connected. I'll come to this later.

An important point is that the Royal Family didn't start with Queen Victoria. The present ruling house, the House of Windsor (previously called Saxe-Coburg and Gotha), was founded by her, but the royals can trace their family tree way, way back - through the houses of Hanover, Stuart, Tudor, York, Lancaster etc. etc. to William the Conqueror who invaded England in 1066 and even beyond that - not taking into account their Scottish ancestry here.

You may ask why Victoria "created" a new house - well, her predecessor William IV (the last of the Hanoverians) died without any children, so the crown passed to the first eligible heir, and this was Victoria, his niece. She married a German prince (Prince Albert) who was from the house of Saxe-Coburg and Gotha. Because she was a woman, she couldn't continue the patrilineal line of the Hanoverians (and similarly, the Hanoverians came to the throne because the Stuarts had died out etc. - this is why there has been numerous "houses" through whom the royal blood has continued, but not always in a straightforward manner). In the wake of WWI, king George V got rid of all his German titles (which he inherited as the ancestor of Prince Albert) and changed the name of the royal house to Windsor, which remains to this day.

Moving on to Princes and Princesses. In a monarchy, there is always one single monarch (the sovereign). If he is a man, he is called King and his wife is known as Queen. If the sovereign is a woman (as is the case now), she is called Queen but her husband is not called King, he is called Prince Consort instead. This is why Elizabeth's husband is Prince Philip. Essentially the monarchy (and the nobility) are very male-centric and patrilineal in nature, so when the bearer of a title is male, his wife is essentially known as the female equivalent of his title; but if the bearer of a title is female, her husband is afforded none of the privileges.

So, the King and Queen (or the Queen and Prince Consort) have children and they are called Princes and Princesses. You will see that Elizabeth's children are Prince Charles, Princess Anne, Prince Andrew and Prince Edward. Now the first-born princess customarily gets the title "Princess Royal" (the current bearer is Princess Anne) but this is not automatic and has to be approved by the monarch. There can also be only one Princess Royal alive at any time. For example, the current queen was never the Princess Royal, because at the time the title belonged to her aunt. By the time she died, Elizabeth had already become Queen.

Similarly, the heir apparent (first-born), currently Prince Charles, is given the title Prince of Wales. His wife is then of course Princess of Wales. The current wife of the Prince of Wales does not use this title (out of respect for his first wife Diana) even though she technically is the Princess of Wales, and is known as the Duchess of Cornwall instead (Duke of Cornwall is one of many titles of Prince Charles).

In the male-centric fashion, the wives of Princes become Princesses. But they are princesses by marriage (because their husbands are Princes), not princesses by birth (like Princess Anne) and this is an important difference. When Prince Charles married Diana, she became Diana, Princess of Wales and not Princess Diana (even though everyone called her that). She could never be Princess Diana because she was not of the blood royal. The same goes for Kate and Meghan recently. The children of Princes are also customarily granted princely status. For example, the sons of Prince Charles are Prince William and Prince Henry; the daughters of Prince Andrew are Princess Eugenie and Princess Beatrice; the children of Prince Edward should be Prince James and Princess Louise, but they were not given this title according to their parents' wishes, so they are using another title of their father. By contrast, the children of Princess Anne don't have any title because their father (who is not a prince) doesn't have any title.

Now, on to other titles. I talked about nobility right at the beginning, so here it comes into play. It is a separate class of titles and can coexist with titles of Prince, with Prince(ss)es being exclusively royal. British nobility is composed of five ranks: Dukes, Marquesses, Earls, Viscounts and Barons. These titles were given to individuals throughout the history and are hereditary (this question is not about nobility so I am simplifying greatly here, feel free to ask more about it), and it is customary for the sovereign to bestow titles (usually the rank of Duke) to his/her family. This has usually been done on the occasion of marriage. Prince William was made Duke of Cambridge on his wedding day, so when he married Kate Middleton, she became the Duchess of Cambridge as well. (Prince Edward's title is the Earl of Wessex, and this is the title his children are using even though they are technically Prince and Princess). These titles of nobility have a complex system of how their children are addressed, known as courtesy titles (for example Earl of Wessex's son is Viscount Severn for example, but his daughter is Lady Louise Windsor). Princes (like George) tend to be known as such, even though they might have other titles. For example Prince William is also Duke of Cambridge, Earl of Strathearn, and Baron Carrickfergus, so Prince George could be using these titles; but given that the status of a Prince is of greater importance, this is what is used.

The whole thing is quite complicated and I could write many more paragraphs on the royal surname, nobility system, knighthoods and baronetcies etc. - I think I have answered all of your questions so far, you're welcome to ask more!