"A woman eats the apple."
Translation:Een vrouw eet de appel.
Eten (to eat) is indeed one of many irregular verbs in Dutch (welcome to Dutch, a language with more exceptions than rules :P (I'm a native speaker)). However, the irregularity is only relevant in the past tense; some additional information can be found here: http://en.wikipedia.org/wiki/Past_tense#Dutch.
WarmFoothills is right about why 'ik' and 'hij/zij/ze' are the same. When writing a verb, you first take the stem (remove the 'en' at the end of the verb and than add:
ik (first person singular): [nothing]
jij/u (second person singular): -t (most of the time, there is one exception)
hij/zij (third person singular): -t
all plural forms: -en (yielding the infinitive)
So, for 'eten', you remove 'en' at the end, so the stem becomes 'et-'; then adding [nothing] or a 't' both result in 'et' and adding 'en' yields 'eten'. This however yields a change of the sound of the word in the singular (not in the plural).
Because the sound of the first 'e' changes, an additional 'e' is needed, so 'et' becomes 'eet'. To elaborate: both the first 'e' in eten and 'ee' in 'eet' are pronounced somewhat like the 'a' in 'late'; 'et' is pronounced like 'wet' without the 'w'. (Perhaps someone with knowledge about IPA could elaborate on this.)
Sorry about the long post, I got a bit carried away :).
'De' and 'het' are the definite articles in Dutch, equal to 'the' in English.
Dutchs nouns have three genders: masculine, feminine and neuter.
'De' is used for masculine and femine nouns in the singular. 'Het' is used for neuter nouns in the singular.
Furthermore, 'de' is used for all plural nouns.
There are some rules (although they are mere guidelines because of the exceptions everywhere in Dutch :P), but you'll have to learn the gender by heart. Some information on these rules can be found here: https://en.wikipedia.org/wiki/Gender_in_Dutch_grammar and here: https://onzetaal.nl/taaladvies/advies/de-het-algemene-regels The last one links to the site of the society dedicated to the Dutch language. It is in Dutch, but it is very useful for some background information.
I most cases, for example komen (to come) is a regular verb.
- Ik kom
- Jij komt
- Hij komt
- Wij komen
- Jullie komen
- Zij komen
Another example of a word of which the stem ends on a -t is zitten (to sit), so you don't get an extra -t.
- Ik zit
- Jij zit
- Hij zit
- Wij zitten
- Jullie zitten
- Zij zitten