Now we're on page 2 - so mentally turn the page. Don't forget to initial the first page.
The Author shall give the Publishers full details of any copyright material (both textual and illustrative) which is not original to the Author. The Publishers shall clear the permissions for such material, and the cost of any permission fees shall be borne by the the Publishers, but if in their opinion any permission fees are excessively high they can decline to include the relevant copyright material in the Work.
This means that you can't stick bits of someone else's book or other copyright work in your book without telling the publisher all about it. The publisher will pay the costs involved, but can refuse to include the material if it costs too much.
full details of any copyright material = the title of the work, the name and contact details of the copyright holder, the publisher, the publication date, the ISBN (if it's a book), the reference for the bit you want to use (eg page or line numbers), the text (or picture) you want to use. It's illegal to reproduce copyright material without the permission of the copyright holder, and if you include copyright material and don't tell the publisher, you'll be in trouble if the copyright holder sues. 'Trouble' can be anything up to and including ending up in the bankruptcy courts.
any copyright material = any material covered by copyright; in the UK, copyright generally extends for 70 years after the death of the original author. But there are plenty of special cases and if you are using anything other than a book published relatively recently, you need to check carefully. For instance: you have found some letters written during World War I by a soldier killed at Verdun that have never been published. Copyright in these expires on 31st December 2039. Now suppose your soldier survived the war and died of old age in 1984, still not having published his letters. Copyright in these expires 70 years after the death of the soldier, in 2054. But if the soldier was killed in an air raid in the World War II, copyright would still expire on 31st December 2039. Confused? Here's a useful flowchart to help you work out whether something is still in copyright.
Don't assume that because something was written a very long time ago it is out of copyright. Milton's Paradise Lost was first published in 1667, but that doesn't mean you can grab your copy of Paradise Lost and use as much of it as you like. There is also copyright in an edition - so the punctuation, corrections, capitalisation, page layout and so on are copyright even though the actual words are not. If you want to quote from something old, you can easily find an old edition that is out of copyright or an online edition that is copyright free. There is also copyright in translations, so if you want to quote a chunk of Goethe, that will be covered by copyright restriction unless you choose a very old translation.
If you are writing fiction, copyright material you are most likely to use would be any snippets of songs that your characters sing, any quotations from other books, or from poems, films, TV series and so on. If you are writing non-fiction, it's more likely to be obvious to you that you have used copyright material as it will probably be chunks quoted from another book.
In the UK, there is no legal limit set on how much of a work you can quote without permission, and some copyright holders are very vigorous in pursuing people they consider have breached their copyright. T.S. Eliot's widow is notorious for this and there are tales of her objecting to quotations of more than two consecutive words of Eliot's poetry without permission. Some reproduction of copyright material is allowed as long at is it considered 'fair dealing' or 'fair use'. Quotation for the purposes of 'criticism or review' is allowed under this provision as long as the quotation is not excessive and the source is acknowledged, so a book of literary criticism will not need clearance for every single quotation. (If you are publishing such a book, the publisher should know what they are happy quoting without permission.) Other things covered by fair dealing include news reporting and incidental use (eg accidentally showing a copyright poster in the background of a photograph). There is a good summary of fair dealing here.
The Publishers shall clear the permissions = you give the publisher the details of what you are using, it's up to them to get permission to use the material. Look out for this clause, as some publishers will try to make it your responsibility to clear permissions. The publisher should have staff who know how to clear permissions and do it routinely. Let them do it. You might make an exception if, for instance, the copyright holder of unpublished letters is your great aunt. Otherwise, let the publisher do it.
the cost of any permission fees = the amount it will cost to use the material
Often, permission to use copyright material is given in exchange for payment of a fee. It is entirely up to the copyright holder how much they want to charge, but it usually relates to the amount that is being reproduced and the use it is put to. So the fee is likely to be lower for a book aimed at the educational market, printing 5,000 copies and selling into schools and libraries, than for a glossy coffee-table book that will sell 50,000 copies at £25 a time. Large publishers and picture agencies usually have an established scale of charges. For pictures, the fee varies depending on how large the picture will be when reproduced, whether it is on the cover or internal pages, and whether it is in black and white or colour.
Look out for this clause - some publishers try to make you pay the permission fee! This is especially the case in academic works, and you may not be able to get it changed. I once spotted it in a contract for a book which was to consist of 40% quotation, so it would not have been worth writing the book if I had not had the clause removed as all the income would have gone on permissions clearance.
if in their opinion any permission fees are excessively high they can decline to include the relevant copyright material in the Work = if it costs too much, the material won't go in the book. You might be able to find cheaper replacement material, or you might just have to miss it out. (If it is lines from a song, you might do better to make up a song of your own.)
You need to take good notice of this clause as there will be a later one, usually labelled 'warranty', which makes YOU liable for any financial loss that results from you including copyright material that has not been cleared.