The extraordinary variety of bets regularly placed in Britain and Ireland on the outcome of horse races is a testament to several century's fermentation of greed, hope and invention in the minds of the nation's punters and bookmakers.
The different types of bets explained below are principally the preserve of the off-course or betting shop bookie.
 

Win: Betting on a horse to come first. With some bookmakers, it is possible to specify that your bet be settled on the "first past the post" basis, and will not be affected by any subsequent demotion or disqualification of your selection. Of course this also means that, regardless of the official result, you will only get paid if your horse is actually "first past the post". Some of the independent bookmakers offer the special concession of paying on both the official and the "first past the post" result.

A £5 win bet on horse A - Total stake £5 +tax

Each Way: This is actually two bets: one bet on a horse to win, and one bet on a horse to be placed, literally a bet on a horse in "each way". It is the established practice of British bookmakers to refuse a bet on a horse to be placed unless it is coupled with a win bet in this manner.
A successful win bet is settled in the normal manner, while a  successful place bet will be settled at a predetermined fraction of the horse's win odds. For these purposes, whether a horse can be regarded as achieving a place, and the fraction of the win odds paid, are dependent on the individual bookmaker's rules for a particular race, but the generally accepted definitions are as follows: 

CSF or Computer Straight Forecast: a bet on picking the winner and second-place horse in the correct order.
The returns on a succesful bet are calculated using a mathematical formula which has been established as an industry standard by the leading bookmakers, and, some have suggested, conspicuously weighted in their favour. The fact is that any formula for assessing the chance of a particular "forecast" event that works with the starting prices as its estimation of each horse's merits ( ie already containing a bookmaker's percentage ) will always favour the layer.
Many commentators cheerfully admit that the bookmakers' formula is operating at a higher level of numeracy than they or many other punters possess, an admission which has yet to prompt a series of remedial classes sponsored by Ladbrokes.

A £ 5  CSF on horse A and horse B is one bet - horse A  to finish first and horse B to finish second, total stake £5 +tax
A £ 5 reverse CSF on horse A and horse B is two distinct bets - horse A to finish first and horse B to finish second, and horse B to finish first and horse A to finish second - Total stake £10 +tax.

Most bookmakers will accept bets at Tote dividend "prices" and many have facilities for betting directly into the pools via Tote Direct, but the above three bets are the basics of British and irish punting.

It is in readily accepting multiple bets, combinations of the above bets on two or more events, that the High Street bookmaker both exceeds the service usually offered by the on-course bookie, and hopes to pay for little luxuries like international leisure conglomerates.

Double: A combination of two bets in different races. If the first bet is succesful then the winnings and stake are used as the stake on the second bet. If either bet is unsuccessful then there are no returns on the double.

A £5 each-way double on horse A and horse B = a £5 win double A+B, and a £5 place double A+B. Total stake £10 +tax.

If both horses in this example were to win 16 runner handicaps, each at odds of 6/1, then the returns would be worked out as follows:
£5 win A at 6/1=£35 ( £30 winnings +£5 stake returned ) then
£35 win B at 6/1 =£245 (£210 winnings + £35 stake returned )
The place double is entirely seperate:
£5 place A at 6/4=£12.50 (£7.50 winnings + £5 stake returned ) then
£12.50 place B at 6/4=£31.25 (£18.75 winnings + £12.50 stake returned
Total return on the double = £276.25

Treble: Three bets in three different races. As with the double, and with all cumulative bets, if any of the constituent bets are unsuccessful, then there are no returns.

Trixie: Three selections combined in one treble and three doubles. Four bets in all.

Patent: Three selections combined in one treble, three doubles and three singles. Seven bets in all.

Yankee: Four selections combined in one fourfold ( four horse accumulator ), four trebles and six doubles. Eleven bets in all.

Lucky 15: Four selections combined in one fourfold, four trebles, six doubles and four singles. Fitfteen bets in all. The "lucky" element is a series of concessions offered by bookmakers eager to solicit multiple bets. Individual bookmakers' rules vary, but the bonuses and consolations are as follows:
Only one winner: double the winner's odds paid.
All selections are winners: 10% of winnings bonus paid.

Canadian or Super Yankee: Five selections combined in one fivefold, five fourfolds, ten trebles and ten doubles. Twenty-six bets in all.

Lucky 31: Five selections combined in one fivefold, five fourfolds, ten trebles, ten doubles and five singles. Thirty-one bets in all. Same bonuses / consolations as Lucky 15.

Heinz: Six selections combined in one sixfold, six fivefolds, fifteen fourfolds, twenty trebles and fifteen doubles. Fifty-seven bets in all.

Super Heinz: Seven selections combined in one sevenfold, seven sixfolds, twenty-one fivefolds, thirty-five fourfolds, thirty-five trebles and twenty doubles. One hundred and twenty bets in all.

Goliath: Eight selections combined in one eightfold, eight sevenfolds, twenty-eight sixfolds, fifty-six fivefolds, seventy fourfolds, fifty-six trebles and twenty-eight doubles. two hundred and forty-seven bets in all.

Union Jack: Nine selections, numbered one to nine, and combined in eight trebles. The banker or best bet selection is numbered five, and forms the centre of a "union jack" arrangement:

the trebles are as follows: 1-2-3, 1-4-7, 1-5-9, 2-5-8, 3-5-7, 3-6-9, 4-5-6, 7-8-9.

Any-to-come or A-T-C: An instruction that frees the punter from trotting backwards and forwards to the betting shop while out shopping on a Saturday - a whole afternoon's punting can be outlined in one morning visit to the bookies, and there's no risk of being swept away by the heady atmosphere of your local layer's and having more on than you can afford.

It is possible to list a series of bets, and specify that the second is only to be made if there is "any to come" ie any returns, from the first; the third to be made only if there is "any to come" from the second etc. Accumulator's for the cautious, a-t-c bets allow the punter to specify the portion of a particular bet's returns to have on the next selection. For example:

£10 win horse A, a-t-c £5 win horse B, a-t-c £2.50 e/w dble horse C + horse D.

It's possible, therefore, for A to win and B to lose (in which case there's "nothing to come"  for the each-way double - the bet is not placed) and the punter will receive returns on £10 win A minus £5.
A-T-C has been touted as a system for betting tax-free, in that you only pay tax on the stake for your first selection.
A-T-C bets are just the thing to annoy your bookmaker and baffle the junior settler.

Up and Down or Single Stakes About: Two selections in two single bets with an additional A-T-C proviso on each bet. eg
£10 Up and Down horse A and horse B: £10 win A a-t-c £10 win B; £10 win B a-t-c £10 win A. Total stake  £20 +tax

There are numerous local names for and variants of the above, but most bookmakers will accept these bets as described.