Gerund mood
In this post I will try to make a recap of all the past post I've written on this subject, and possibily add some more examples.
The gerund mood only has two tenses: present and past. You form it by adding a different suffix according on the verbs' conjugation you have to use. 1st conjugation (verb's suffix: -are, e.g. cantare) : -ando (cantando) 2nd conjugation (verb's suffix: -ere, e.g. vedere) : -endo (vedendo) 3rd conjugation (verb's suffix: -ire, e.g. uscire) : -endo (uscendo)
There isn't always a specific subject with it, but in general, it works together with the main sentence's subject (when not differently explicited). It can also be used impersonally. It is never conjugated accordingly to genders/numbers of the subject, as it is fixed.*
e.g. (io) Cammino cantando = I walk (while) singing (while I sing/I am singing).
*If anything, it is the past participle that comes with it at the past tense, that is conjugated, especially if you also have a personal pronoun together with the verb: e.g. mangiando = eating; mangiandoli = eating them avendo mangiato = having eaten; avendoli mangiati = having eaten them (let's suppose "li" stands for ravioli or any other masculine and plural dish/food you have mentioned before in your speech- check the personal pronouns post if you need or leave an ask).
At the present, it is used to stress (differently from the present simple of the indicative mood, that refers to a time that is present but not too specific)** the temporal value of an action, and the contemporaneity with the main clause. Ofc, that is not the only value it has (a present tense gerund may be used in a conditional, causative, modal and concessive way too).
e.g. Scrivo ascoltando musica = I write while I am listening to music.
At the past, it tells us about an action that happened before the main clause's one. Once again, the temporal value isn't the only way to use this mood's tense.
e.g. Avendo terminato il lavoro, tornò a casa =After he finished his/her job, s/he went back home.
When you find it together with "stare" or "andare", you have the progressive gerund: it means you are telling about an action that is happening exactly in this moment.
e.g. (in questo momento) Sto scrivendo questo post = (right now) I'm writing this post.
** Let's talk a little more about the tense consistency and the difference with the present simple of the indicative mood. Let's suppose you are writing a letter while eating an apple.
Present simple, indicative mood: - Scrivo una lettera e mangio una mela (it doesn't mean you're doing those two things now and together: I may also think you are going to write a letter, then eat the apple and then do something else). - Scrivo una lettera mentre mangio (=mangiando) una mela. (it doesn't mean you're doing those actions right now, but I might suppose so. It could also be that you usually act this way: when you write letters you like to eat apples. Using "mentre + indicative" or the gerund, I know you do them together.)
Progressive gerund: - Sto scrivendo una lettera e mangiando una mela = Sto scrivendo una lettera mentre mangio una mela. (it means you're doing those actions together and now, exactly as in the English "I'm writing a letter and/while eating an apple".)
Always watch/think about the main clause's verb first, then decide, according on your needs, which tense is more appropriate. If you're doing something right now, use the progressive gerund, if the action isn't happening exactly now or you don't feel like stressing that you are doing that thing now (it may be not necessary), go with the present simple indicative mood.
[please for more examples, check: the gerund mood guide + gerundio progressive gerund verbs moods/tenses participio vs gerundio*]