"Spaniard" is a noun, but it is slightly old-fashioned. It refers to ONLY to a male person of Spanish nationality. You wouldn't call a Spanish woman or a child a "Spaniard".
"Spanish" is an adjective. It can refer to a man, woman or child of Spanish nationality, or anything else in or originating in Spain: the language, a place, an object, an institution...
As with any nationality adjective, you can also use "Spanish" as a plural noun to refer a group of people (or the whole nation) by preceding it with "the". For example, "The Spanish won the game" : a perfectly correct sentence.
"Spanish" can't be a singular noun, however. This is the same for all nationality adjectives ending in "sh", "ch" and "ese". So you can't say "He's a Spanish" - you have to say "He's Spanish" or 'He's a Spanish man".
This is different from nationality adjectives ending in "an" ( such as Brazilian) which can be both an adjective and a noun.
To be honest, I wouldn't worry too much about the word "Spaniard". It's a far less useful word than "Spanish", and ( in common with many demonyms such as "Scotsman' ) may soon become outdated.
These days, people very rarely refer to themselves or to people they know as Spaniards. We tend to prefer the adjective form. If you met someone on a flight from Madrid, you would not say "Are you a Spaniard?" - you'd say "Are you Spanish?".
He is Spanish. (Adjective.)
He is a Spaniard. (Noun.)
@Suki has made some very good points, and she is mostly right. "Spaniard" is a little antiquated. There is one point she makes about "Spaniard" not denoting a Spanish woman, that I am not sure about. (Personally I have never heard this appelation to a Spanish woman). Dimly, I remember, not having "Spaniard" for her, does English have a special word for "Spanish woman" (excepting "senora" etc)? Do you know, Suki? Big thanks!
@Aliph. You are right, "Spanish" must be used in the plural. Let's say Jose is Spanish. "Is there a Spanish person here?" "Yes, we have a Spanish person (a Spanish??)", or "yes, we have a Spaniard". "Are any Spanish people here?" "Yes, there are Spanish people here". "Are any Spaniards here?". "Yes, there are (Spanish??)/Spaniards." "Did you see Jose and Teresa?" "Yes, I saw 2 Spaniards in the room", "yes, I saw 2 Spanish persons (2 Spanish??) in the room.
@Suki, are the "??"s good English, please? Although they sound modern, they sound weird to me. Thanks.
The Oxford-Hachette English-French Dictionary says that Spanish ( plural) is a noun .
So does the Oxford dictionary on my IPad . The Spanish (plural) = the people of Spain.
I agree with @Janine. She uses the best, and proper, English.
"The Spanish will win the game", "the Spanish living in Portugal", ... you often hear among Brits. It is colloquial, but, as she says, should be "the Spaniards will win..", "the Spaniards living in..", etc.