Sian Higgins of Merthyr, South Wales, writes Partially thanks to Amanda Cassidy I have the following information about William and Eliza Sells (sometimes referred to as Sills)
1851 census index for Redbourn: Blackhorse - Sells - Charles age 7m, Eliza age 21 and William age 25.
1881 census for Redbourn: William Sells,56 Beerseller, Church End (born Luton, Beds), Eliza 51 (Born Redbourn), Henry 10, Alfred 9, Olive 7.
1891 census for Redbourn: Sills - Eliza age 61, Olive age 18 and William age 66, a Publican, living at The Jolly Gardener.

I have details of christenings of the following children from the Vital Records Index:
Charles 1850, Sarah 1852, Jan 1853, Walter 1855, Eliza 1857 all in Redbourn- and know of the following Mary Ann b. c1863,Henry b. c1871, Alfred b. c1872, Olive b. c 1874

Trade directories under Redbourn:
1854 Craven and Co's: Sells, William, Victualler, "Black Horse"
1864 Cassey - Sells, William, Black Horse Inn
1870 Kelly's - Sells, William - Black Horse
1882 Kelly's - Sills, William, Beer Retailer
1890 Kelly's - Sills, William, Beer Retailer
1902 Kelly's - No Sells or Sills listed for Redbourn.

Do you have any history of the Black Horse Inn or the Jolly Gardener? 

I have no real information on these. The order properties were visited in Redbourn in the 1851 census (see Locating Census Addresses on Maps) suggests the the Black Horse was in Lamb Lane which ran from the north end of the High Street towards Church End and Hemel Hempstead. It was almost certainly not where the modern Black Horse Lane is - as this area was fields in circa 1880 (see Old maps of  Hertfordshire) although perhaps not far away. It might even be that the Black Horse and the Jolly Gardener were one and the same .... You need an answer from someone with more local historical knowledge.

I have added a page on Redbourn and another Coaches and Inns, Redbourn, 1830's

Do you know if there is any licensing information that may be available? 

Any official information will be held at HALS, and I have no experience of these records. There is also a lot of information in local newspapers, such as the Herts Advertiser, which not only reported licensing hearings, but also reported on criminal cases - a significant number of which seem to mention the pub where someone had been drinking, where stolen goods were exchanged, etc., etc. Unfortunately these records are not indexed and you could wade through many years papers, week by week, without finding a reference to the specific pub you are interested in.

I am about to look at the St Catherines [GRO] Index to find William and Eliza's marriage.  I understand from Jane's birth certificate that Eliza's maiden name was Arbrough.  Is that a common name in Herts as I found no reference to it on the IGI for Herts?

I don't know the surname at all - it is not in the 1881 census either - so I would suspect a spelling error. See A Comedy of Errors for the causes of errors in the official records. As they probably married in Redbourn I suggest you try the Redbourn marriage register (which as far as I know has not yet been indexed by the LDS) as, if this is where they are married, this will give you the original register entries - with the individual signatures. Try HALS for certificates for events that happened in Hertfordshire.

Please could you suggest any other sources that might provide information about publicans so I can pad out my research.

 Hertfordshire Inns & Public Houses, by Graham Jolliffe & Arthur Jones, covers establishments which still existed at the time the book was researched and the only reference is a note that the Jolly Gardeners closed down sometime between 1937 and 1962. According to The Hertfordshire Village Book "The Jolly Gardeners, once an inn and now a private dwelling, is the oldest house in the village" so you may be out of luck - although if the building is that old it may well be recorded in records relating to listed buildings.