Hinna is a very useful, very common Swedish word, which can be translated as 'to have the time to' or 'to make it/to manage [to do something]'.
It's often used as a modal or 'help' verb, meaning you'll usually see it followed by another verb, such as in the following examples: Hinner du hämta maten? (Do you have time to pick up the food?), tåget går snart så vi hinner inte köpa en bulle (the train is leaving soon so we don't have time to buy a pastry), or tyvärr hann han inte göra läxorna (unfortunately he didn't have time to do the homework – hinna is an irregular verb so the past tense is hann).
You'll still see and hear hinna used as the only verb in the sentence, in which case another verb is implied. This is often the case in response to a question, for example: Kommer du till festen i kväll? Nej, jag hinner inte (Are you coming to the party tonight? No, I don't have the time). Note that it's still correct Swedish to say jag har inte tid, a literal translation of 'I don't have time', but using hinna correctly is sure to get you brownie points.
Hinna is usually used either with a negative (such as aldrig or inte), or in a question or speculative tense (such as jag hoppas vi hinner! or 'I hope we'll make it in time!'). Basically, you'll hear it whenever there's a question mark over whether something's going to be possible; if you say jag hann göra det (I managed to do it in time), you're implying that there was a doubt as to whether you'd be able to.
Hinna can also be combined with a variety of prepositions, which tweak the meaning slightly.
You can use hinna med if you want to follow it with a noun rather than a verb. So you'd say: jag hinner inte svara på alla frågorna (I won't manage to answer all the questions), but jag hinner inte med alla frågorna (I won't have time for all the questions),and you'd say jag hann inte äta lunch (I didn't have time to eat lunch) but jag hann inte med lunch (I didn't have time for lunch).
Hinna upp means 'to catch up with/to', for example hon sprang för att hinna upp bollen (she ran to catch up to the ball).
Hinna ikapp and hinna ifatt both have the same meaning of 'to catch up', but usually refer to abstract concepts rather than physically catching up to something in speed. For example, jag talar inte svenska så jag behöver tid för att hinna ikapp (I don't speak Swedish so I need time to catch up), or de fattigaste städerna behöver hjälp för att hinna ifatt de rikaste (the poorer towns need help to catch up with the richest).
And hinna fram till means something like 'to make it [in time]' and is often used with geographical locations: for example: jag hoppas att jag hinner fram till sjukhuset innan förlossningen (I hope I make it to the hospital before the birth). You can also use it in the abstract sense, such as vi hinner tyvärr inte fram till denna frågan ikväll (we won't get to this issue tonight).
According to the Swedish Academy's dictionary, hinna comes from an Old Norse word inna, meaning to 'reach an end', and is also related to old Germanic words for 'catch' and 'prey', as well as English 'hunt'. This helps to explain how it can be used to talk about concluding a task both in terms of geographical movement and movement through time.
Finally, there's also a Swedish noun, hinna, which refers to a membrane, film, or thin surface and is used in compound nouns such as trumhinna (eardrum). Unless you work in certain scientific fields, you're far more likely to come across the verb than this noun.
Jag hann aldrig säga att jag älskade henne
I never had time to say I loved her
Det finns alltid för mycket att göra. Jag hinner inte!
There's always so much to do. I don't have time [to do it all]!