Is the "Roman alphabet" what we use for English?

The term for the name of the script is Latin, at least according to Unicode.

U+0041 ‭ A  LATIN CAPITAL LETTER A
U+0042 ‭ B  LATIN CAPITAL LETTER B
U+0043 ‭ C  LATIN CAPITAL LETTER C
U+0066 ‭ f  LATIN SMALL LETTER F
U+0067 ‭ g  LATIN SMALL LETTER G
U+0068 ‭ h  LATIN SMALL LETTER H

U+00DE ‭ Þ  LATIN CAPITAL LETTER THORN
U+00DF ‭ ß  LATIN SMALL LETTER SHARP S
U+00E0 ‭ à  LATIN SMALL LETTER A WITH GRAVE 
U+01BF ‭ ƿ  LATIN LETTER WYNN
U+021D ‭ ȝ  LATIN SMALL LETTER YOGH

Contrast those with non-Latin letters, like these:

U+0393 ‭ Γ  GREEK CAPITAL LETTER GAMMA
U+0394 ‭ Δ  GREEK CAPITAL LETTER DELTA
U+03B4 ‭ δ  GREEK SMALL LETTER DELTA
U+03B5 ‭ ε  GREEK SMALL LETTER EPSILON
U+03B6 ‭ ζ  GREEK SMALL LETTER ZETA

U+0416 ‭ Ж  CYRILLIC CAPITAL LETTER ZHE
U+0417 ‭ З  CYRILLIC CAPITAL LETTER ZE
U+0418 ‭ И  CYRILLIC CAPITAL LETTER I

Any code point with the Unicode character property Script=Latin is deemed a Latin letter — presuming it is a letter. There are a few Latin code points that count as numbers, too, like

U+2180 ‭ ↀ  ROMAN NUMERAL ONE THOUSAND C D
U+2181 ‭ ↁ  ROMAN NUMERAL FIVE THOUSAND
U+2182 ‭ ↂ  ROMAN NUMERAL TEN THOUSAND

Bringhurst in Elements of Typographic Style talks about how the Latin alphabet is not the ASCII alphabet, but contains in fact hundreds more glyphs, or sorts as a typesetter might say. Most of these we only use in English when writing an unassimilated loan words, or someone’s name, if at all.

U+00E6 ‭ æ  LATIN SMALL LETTER AE
U+00E7 ‭ ç  LATIN SMALL LETTER C WITH CEDILLA
U+00F0 ‭ ð  LATIN SMALL LETTER ETH
U+00F1 ‭ ñ  LATIN SMALL LETTER N WITH TILDE
U+00F8 ‭ ø  LATIN SMALL LETTER O WITH STROKE

And it gets weirder, because as more world languages have adopted the Latin alphabet, they often have reason to add new letters to it for their own needs.

U+01A6 ‭ Ʀ  LATIN LETTER YR
U+0238 ‭ ȸ  LATIN SMALL LETTER DB DIGRAPH
U+0242 ‭ ɂ  LATIN SMALL LETTER GLOTTAL STOP
U+0259 ‭ ə  LATIN SMALL LETTER SCHWA
U+1D24 ‭ ᴤ  LATIN LETTER VOICED LARYNGEAL SPIRANT
U+1D79 ‭ ᵹ  LATIN SMALL LETTER INSULAR G
U+0298 ‭ ʘ  LATIN LETTER BILABIAL CLICK

"Roman" is a type of lettering based on the letter-forms adopted in Ancient (Imperial) Rome:

  ^{[Image credit: Me]}

The name is commonly used for the writing of a number of languages, as you suggest.

Different languages may have different alphabets. For example, Welsh includes DD and LL as entities separate from D and L, as well as a few other digraphs, and does not include K, Q or Z. However the letter-forms are called "Roman", as illustrated here.