"Di" usually we can translate in english like "of": his role is specify something, a characteristic or possession.
With "Da" we can think to "from": means the provenience of a movement, of a time o stuff like this.
These are PREPOSIZIONI SEMPLICI, if we add to an article we would have PREPOSIZIONI ARTICOLATE (means prepositions + articles).
"Della" is DI+LA, "Dalla" is DA+LA. So you can use them when you need to express a subject of your complement singular and female (article "la").
Examples:
"La casa DI Lucia" - "Il tavolo DI legno"
"Da questa strada vedo il mare" - "Il negozio è aperto DA martedì A giovedì"
"La casa DELLA zia" - "DALLA vacanza in Francia ho portato del formaggio"
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I hope to be usefull, if you have any question write me a message.
Have a nice day!
Rolando