Depends which kit it is, the basic one or the full 'starter kit'
The full starter kit should include everything you need ( possibly with the exception of a keg or bottles to put the finished brew in ) the basic one doesn't.
It should say on the box or in the instructions what you'll need.
If the kit doesn't include them, at the very least you'll need : Something to do the main ( primary ) fermentation in, something - bottles or a proper brewing keg - to put it in afterwards ( secondary fermentation ) something to transfer it from one to the other, possibly some sugar to add to the initial brew & some to add to the bottles/keg for the secondary fermentation ( unless you like flat beer
) something to sterilise everything with & something to stir the initial brew with.
If you've got something like a Wilko store nearby, you should find everything you need there.
What you get depends on whether it'll be a one-off or if you plan to make more later.
Assuming it's a one-off :
Primary fermentation -- a lidded 25L or larger plastic bucket...you can use a proper fermentation bucket, or you could take the - possibly cheaper - option of using one of the large storage buckets DIY shops sell ( for paint, animal feed, etc. )....the good news is that - unlike wine - it doesn't need to be airtight for primary fermentation of beer.
You'll need to stir the mix when you first stick it in the bucket, but you shouldn't need to do so after that....they recommend using a plastic spoon, but you can get away with using a large metal one.
Secondary fermentation -- some people prefer kegs, others prefer bottles.....bottles are cheaper, in fact they could even be free if you clean & re-use old ones which have contained fizzy drinks or water. Beware of using the flimsy ones used for some still bottled water, though, as they won't withstand the pressure. You can even re-use the screw-tops, just make sure they really are screwed down tight or you'll end up with flat beer.
Cleaning -- you can get sterilising fluid specifically marketed for use in brewing....a bit of a cheat, though, is to use Miltons Fluid ( used for sterilising babies bottles, etc ) which will also do the job & has the added bonus of not needing rinsing out after use ( theoretically you can get away with just using boiling water, but that's up to the individual )
Transferring -- you can buy a proper syphon for a couple of quid....when you get it, you'll find that it's essentially a 6' length of clear flexible plastic tubing ( usually about a 4-5mm bore, so you could just go to a DIY store & buy some of that. Alternatively, if you're careful you can get away with using a plastic or glass measuring jug.....either way, you'd need to avoid stirring up the sediment at the bottom of the bucket.
What this means is that with a 23L kit you'll usually have to leave about 2L in the bottom of the fermentation bucket ( I won't mention coffee filter papers & plastic funnels
Sugar -- some kits need you to add sugar for the primary fermentation, others don't. If you can't see it listed as 'brewing sugar' in a shop, just look for dextrose ( it's the same thing ) although you can get away with using ordinary sugar. For secondary fermentation, you'll only need a small amount ( about a flat teaspoon-full for every 500ml )....put the sugar in the bottle before adding the beer.
Think that's about it.
You could also add a hydrometer & a thermometer - the latter's more important as the yeast only works properly between certain temperatures ( usually somewhere around 18-28C )....too hot & you could kill it, too cold & it'll take a lot longer to work.