universally / on all sides
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universally
By everyone or among everyone; generally. Expresses that something is widely known, widespread, or recognized.
Die neue Lehrerin ist bei Schülern und Kollegen allseits beliebt.
The new teacher is universally popular with students and colleagues.
The adverb 'allseits' (universally / on all sides) is made up of 'all' (all) and the ending '-seits' (on the side of). You can recognize in it the structure that also appears in words like 'einerseits' (on the one hand) or 'beiderseits' (on both sides).
Die Entscheidung wurde allseits begrüßt.
The decision was universally welcomed.
'Allseits' (universally / on all sides) belongs more to written or formal language. It is used less often in everyday conversation.
Der Professor war für seine Forschungen allseits bekannt und geachtet.
The professor was universally known and respected for his research.
You can recognize the figurative meaning ('bei allen' — among everyone) from verbs or adjectives that express an opinion, e.g. beliebt (popular), bekannt (known). You can recognize the spatial meaning ('von allen Seiten' — from all sides) from words that describe a position in space, e.g. umgeben (surrounded), Wald (forest).
Ihre Hilfsbereitschaft ist allseits bekannt.
Her helpfulness is universally known.
In the meaning 'bei allen' (among everyone), 'allseits' (universally) is often combined with adjectives such as 'beliebt' (popular), 'bekannt' (known), 'geachtet' (esteemed), or 'respektiert' (respected).
Der scheidende Direktor war bei den Mitarbeitern allseits respektiert.
The outgoing director was universally respected by the employees.
everywhere
'überall' and 'allseits' can both express that something does not occur in just one place. 'überall' is the normal everyday word for 'in all places'. 'allseits' sounds more formal and often means 'among everyone' or 'from all sides'. For example, you can say that someone is 'allseits beliebt' (universally popular) or 'allseits bekannt' (widely known). Spatially, 'allseits' can also mean 'on all sides', but in everyday language 'überall' is usually more natural. The words can be confused because both are connected to 'all'. For places in normal conversation, 'überall' is almost always the right choice.
Frau Becker ist im Verein allseits beliebt.
Mrs. Becker is universally popular at the club.
Nach dem Basteln lagen überall kleine Papiere auf dem Tisch.
After the craft activity, little pieces of paper were lying everywhere on the table.