You need a broker - that is someone who buys the shares for you and usually holds them in an account for you.
Assuming it is a UK company with "plc" in its name, it is pretty easy to find and use a broker. If it doesn't have plc in the name and it is a UK company, then it is probably private - not listed on a stock exchange - and you can't buy shares in it. If it is a foreign company, some of them have listings in London, otherwise you have to ask your broker if they will let you buy American, French , Norwegian etc shares.
You could ask your bank as androbbob suggests. Or you could go to one of the big online brokers: Selftrade and TD Waterhouse are popular. They are both 'cheap' - they are execution-only ie you tell them what to buy, they say that will be £2.35 per share, you say OK, they buy the shares. You don't need the much more expensive advisory brokers for one purchase unless you are buying a lot of something unusual.
You need to ask what your broker will charge to buy and sell shares, whether this changes by how many shares you are buying, whether there are any charges per quarter on your account.
You need to decide if you want to buy the shares in a 'dealing account' or an ISA. Shares in an ISA are tax free - you dont pay interest on dividends or capital gains and you don't need to keep records for tax purposes. If you haven't used up your ISA allowance for the year, an ISA is usually the best choice but check what the charges for ISAs and dealing accounts are. If you have used your maximum £5760 cash ISA this year then you can still put £5760 into a shares ISA, see www.which.co.uk/money/savings-and-investments/guides/stocks-and-shares-isas-explained/what-is-a-stocks-and-shares-isa/
Can I be boring and ask why you want to buy shares in a company? If it's just a bit of money "for fun" in a company your daughter / neighbour / taxi driver says will do well, that's fine if you can afford to lose all the money.
But if you want to start investing in shares and this is just your first step (well done for wanting to take it!), then it might be better to start by buying either a 'tracker fund' or an ETF - that way your money is buying a whole basket of shares, so if one of them does badly, you won't lose a lot of your money. And these tend to be much cheaper than buying unit trusts or investment trusts, which also are 'baskets'.