Latin words must agree in number, person, and gender. In a language like English, where objects don't have genders*, this is an almost entirely alien concept, although when someone uses the adjective "handsome" to describe a female or "pretty" to describe a male, the listener notices this is an unusual gender alignment. But on the whole, in English, we are limited to agreement between singular/plural subject and singular/plural verb.

Verbs and Subjects

Nouns and Verbs
In the following Latin sentence, we see agreement of verb with the number and person of the subject (in the nominative case). The first (numbered) line contains the Latin. The next line shows the English equivalent word-for-word and notes about its grammatical use. The third line contains an English translation of the Latin. The fourth line attributes the quotation, where relevant.

1. omnia_________________ vincit______________ amor
   everything-acc._pl._neut. conquers-3d_pers._sg. love-nom._sg._masc.
   'Love conquers all.'
   (This saying comes from Vergil.)
   
   Remember the subject does not have to come first in Latin. Also note that the subject is 'love' and the verb is 'conquers'. 'All' is the object. Since the object does not affect the ending of the verb, only the word for the subject 'love' matters, in terms of agreement, even though it comes after the verb.
   
   The verb is 3rd person singular and the subject noun (amor) is singular. The verb does not have to agree with the noun in gender.
   If the sentence were "Everything conquers love," the verb would be in the 3rd person plural because 'everything' is plural:
   

1. amorem___________ omnia_______________ vincunt
   love-acc_sg._masc. everything-nom_pl._neut. conquer-3d_pers._pl.
   'Everything conquers love.'
   
   Pronouns and Verbs
   
   In the following sentence, we see agreement of verbs ('hate' and 'fear') with the number and person of the (third person plural) pronominal subjects ('they' for both verbs), which are incorporated in the endings of the verb, so to speak:
   
2. oderint___________ dum metuant
   hate-3d_pers._pl whilefear-3d_pers._pl.
   'Let them hate as long as they fear.'
   (This saying is attributed to Accius).
   The "_nt" is the 3d person plural ending.
   
3. Marmoream______ relinquo______ quam___ latericiam___ accepi
   Marble-acc._sg._f. leave-1st_sg. which-acc._sg._f. brick-acc._sg._f. I took-1st_sg.
   'What I received in brick, I leave in marble.'
   (This saying is attributed to Augustus).
   
   The '-o' on relinquo and the '-i' on accepi show that the verbs are governed by a first person singular subject. Since no noun is specified, the subject is included in the verb and visible in the ending.
   
   Nouns and Adjectives
   Nouns and adjectives must also agree in number. Unlike nouns and verbs, nouns and adjectives must also agree in gender. There are 5 declensions in Latin for nouns, but only 3 for adjectives. The two main ones for adjectives follow the 1st and 2nd noun declensions. This is exceedingly confusing, especially since the second declension for adjectives includes a masculine and neuter form.
   
4. lingua__________ Latina
   language-nom._sg._f. Latin-nom._sg._f.
   'Latin language'
   Latina is the adjective. It has to agree with the noun it modifies, here, lingua.
5. mens____________ sana____________ in corpore________ sano
   mind-nom._sg._f. healthy-nom._sg._f. inbody-abl._sg._neut. healthy-abl._sg._neut.
   'Healthy mind in a healthy body...'
   (This saying comes from Juvenal.)
   
   The same adjective (sana, sano) is used with first a feminine and then a neuter noun. It changes its ending acccording to the gender of the noun and also the case. The first noun is nominative, and the second is in the ablative, so the adjective is also nominative in the first case and ablative in the second case.
   
6. rebus________________ publicis
   things-abl./dat._pl._fem. public-abl./dat._pl._fem.
   'Public things,' i.e., the state
   Here the noun is a fifth declension feminine noun and the ablative/dative ending is '-bus'. The corresponding ending on the adjective is '-is'.

* English has one adjective that differs by gender: blond/blonde. A light-haired man is blond. A light-haired woman is blonde. Blond and blonde are not interchangeable.

Please let me know if I've made an error.

More Latin-English Differences: Agreement | Case | Word Order | Gender | Articles | Alphabet