[Bel-p]

I'm a little confused, I don't think Sodium Carbonate is a salt, but why not?  It's an ionic compound of a metal and a non-metal in the same way that Sodium Nitrate is surely?  Also I think I've seen it referred to as a salt in some places, but is it a salt or not?  Also what about Calcium Carbonate?  Again it's a metal and a non-metal.  Can anyone help me with this?

[AWK]

Salts are formed (formally) from acids and bases in the neutralization reaction
2NaOH + H2CO3 = Na2CO3 + H2O

[Bel-p]

Salts are formed (formally) from acids and bases in the neutralization reaction
2NaOH + H2CO3 = Na2CO3 + H2O

So that's a yes then presumably: Sodium Carbonate (Na2CO3) IS a salt.  
In the equation you've given there is a base (Sodium Hydroxide NaOH) and an acid (Carbonic Acid H2CO3) making a salt and water.   I think our teacher might have said it wasn't a salt so I was confused by that.
I was also confused by the information that an acid and a carbonate makes salt, water & carbon dioxide, as the implication seemed to be that the carbonate was not a salt.  

So what about Calcium Carbonate, CaCO3, presumably that too is a salt?  Let me think this through:
         A base         +     an acid     -->              salt           + water
Calcium Hydroxide + Carbonic Acid -->   Calcium Carbonate + water
       Ca(OH)2        +     H2CO3      -->      CaCO3              + 2H2O

Am I right?

(PS what age is "High School"?  I'm in the UK and am doing GCSE chemistry, usually done at 15-16ish years old, although I'm rather older than that!)

[Albert]

I was also confused by the information that an acid and a carbonate makes salt, water & carbon dioxide, as the implication seemed to be that the carbonate was not a salt.  

2 HCl + CaCO3 -> CaCl2 + H2CO3

However, carbonic acid yields water and CO2 straightaway:

H2CO3 <-> H2O + CO2

So what about Calcium Carbonate, CaCO3, presumably that too is a salt?  Let me think this through:
        A base        +    an acid    -->              salt          + water
Calcium Hydroxide + Carbonic Acid -->  Calcium Carbonate + water
      Ca(OH)2        +    H2CO3      -->      CaCO3              + 2H2O

Am I right?

Yes, exactly.

[AWK]

Carbonic acid exists - it is stable at room temperature (it is true!) but decomposes under catalytic action of water, hence practically does not exist in water solution.

Salts also can be formed by:
metals, oxides, hydroxides in reaction with non-metals, non-metal oxides and acids
Stronger base can replace a weaker one in salt, stronger acid can replace a weaker one in salt, and as a result new salts are formed.
Sometimes, when salt precipitates, even weak acid or base can replace the strong one in the salt.

[UkraineWithAim]

I'm a little confused, I don't think Sodium Carbonate is a salt, but why not?  It's an ionic compound of a metal and a non-metal in the same way that Sodium Nitrate is surely?  Also I think I've seen it referred to as a salt in some places, but is it a salt or not?  Also what about Calcium Carbonate?  Again it's a metal and a non-metal.  Can anyone help me with this?

It is a salt generally, but in some contexts, like Bronsted-Lowry acid-base theory, for example, it can act as a base by accepting a proton:
Na2CO3+2HCl=2NaCl+H2O+CO2
B-L base   Acid
Same with CaCO3

[Bel-p]

Thanks very much everyone, that was really useful.

[hay]

Carbonic acid exists - it is stable at room temperature (it is true!) but decomposes under catalytic action of water, hence practically does not exist in water solution.

Wanna add something fun, no matter how you find carbonic acid, there will always be carbon dioxide Due to the equlibirum nature. Right?