Adam Smith did not used all the terms necessarily in exactly the same way as we use them today, so we have to defer to his explanations in Wealth of Nations.
Nominal and Real Price
Let's start with real and nominal price. First Adam Smith introduces these terms in chapter V which is literally called
Of the Real And Nominal Price of Commodities, or their Price in Labour, and their Price in Money
So this tells you of the bat that real and nominal price are synonyms for price in labour and price in money.
Now that we know these terms are synonyms we can turn our attention to explaining
As Smith further explains in Ch V para. 1:
Every man is rich or poor according to the degree in which he can afford to enjoy the necessaries, conveniencies, and amusements of human life. But after the division of labour has once thoroughly taken place, it is but a very small part of these with which a man’s own labour can supply him. The far greater part of them he must derive from the labour of other people, and he must be rich or poor according to the quantity of that labour which he can command, or which he can afford to purchase. The value of any commodity, therefore, to the person who possesses it, and who means not to use or consume it himself, but to exchange it for other commodities, is equal to the quantity of labour which it enables him to purchase or command. Labour therefore, is the real measure of the exchangeable value of all commodities.
Consequently, according to Smith real price or price in labor is the price you have to pay in for some good. This is based on Smiths labor theory of value (LTV) which posits that labor is ultimately the source of all value. Labor theory of value was in modern economics supplanted by subjective theory of value (as this one can better explain what we observe), so in modern day real price/value has no longer exactly the same meaning and thus beware that nowadays when economists talk about 'real' variables they mean 'inflation adjusted' not their price in terms of labor.
When it comes to nominal price Adam Smith writes in ch V para. 5-7:
Every commodity, besides, is more frequently exchanged for, and thereby compared with, other commodities, than with labour. It is more natural, therefore, to estimate its exchangeable value by the quantity of some other commodity, than by that of the labour which it can produce. The greater part of people, too, understand better what is meant by a quantity of a particular commodity, than by a quantity of labour. The one is a plain palpable object; the other an abstract notion, which though it can be made sufficiently intelligible, is not altogether so natural and obvious.
But when barter ceases, and money has become the common instrument of commerce, every particular commodity is more frequently exchanged for money than for any other commodity. The butcher seldom carries his beef or his mutton to the baker or the brewer, in order to exchange them for bread or for beer; but he carries them to the market, where he exchanges them for money, and afterwards exchanges that money for bread and for beer. The quantity of money which he gets for them regulates, too, the quantity of bread and beer which he can afterwards purchase. ... Hence it comes to pass, that the exchangeable value of every commodity is more frequently estimated by the quantity of money, than by the quantity either of labour or of any other commodity which can be had in exchange for it.
Gold and silver, however, like every other commodity, vary in their value; are sometimes cheaper and sometimes dearer, sometimes of easier and sometimes of more difficult purchase. ... Labour alone, therefore, never varying in its own value, is alone the ultimate and real standard by which the value of all commodities can at all times and places be estimated and compared. It is their real price; money is their nominal price only.
Consequently, as can be clearly seen from the passage alone, Smith considers nominal price as a price of something in terms of money used (at that time gold and silver) as opposed to labor. This usage is close to modern usage since economists still consider nominal price to be price in terms of current value of money, just the contrast with labor is no longer part of the of definition.
Market (actual) Price vs Natural Price
Adam smith only uses the world actual price twice in the whole book and he is very clear on its meaning (see WoN Ch VII para 7):
The actual price at which any commodity is commonly sold, is called its market price. It may either be above, or below, or exactly the same with its natural price.
So market/actual price is just price a good is sold for at the market or market price. This definition overlaps with the definition of nominal price but Smith gives it also different meaning because he contrast it with natural price, which is also a nominal price but a nominal price that should prevail in the long term because as Smith defines it (see WoN ch VII para. 4):
When the price of any commodity is neither more nor less than what is sufficient to pay the rent of the land, the wages of the labour, and the profits of the stock employed in raising, preparing, and bringing it to market, according to their natural rates, the commodity is then sold for what may be called its natural price.
Exchange Price and Consumer Price
There is no occurrence of term consumer price or exchange price in Wealth of Nations, so those are likely mistranslations.
When it comes to consumer price I think that might be a mistranslation of market price, which as already mentioned in the previous section, in chapters where Smith talks about consumer goods.
Exchange price is likely a mistranslation of 'value in exchange'. Value in exchange is concept that Smith introduces because already he notices that labor theory of value is not very good description of reality, since one can exert a lot of labor (e.g. dig a hole and then fill it up) without creating something that can be sold for much. To resolve this discrepancy Smith argued that even though true measurement of value is still labor, there can be additional notions of value in use and value in exchange, and he argued these different notions of value explain the discrepancies between prices and labor cost of producing various goods (although as mentioned in the previous section, this was later discovered not sufficient and rejected by profession during the Marginal Revolution 19th century).