Alert: displays a brief, important message in a way that attracts the user’s attention without interrupting the user’s task.
Unlike popup, more than one alert may be shown at once. Alerts are transient in a different sense than popups: focus is typically not immediately sent to an alert; light dismiss will not clear them, but a timeout may. Typically, alert text is read out dynamically to screen reader users, and so it is recommended that there is little to no interactive content, as this would be included in readout if the entire alert component is a live region (e.g. has ARIA role="alert").
Dialog: a part of an application that a user interacts with to perform a task, for example a dialog box, inspector, or window.
Dialogs and popups are quite similar, but there are some behavioral differences. Whereas only one popup can be displayed at a time (unless there is an ancestry chain), multiple dialogs can be shown at once. Popups are inherently transient, and dismissed upon “light dismiss” interactions. Dialogs are more persistent, and many of these are modal—interaction with the surrounding document is blocked. You might think of the distinction as this: popups are often most appropriate to use in support of the action the user is currently in the middle of, and should keep them in that context. Dialog is often most appropriate if the user should be taken to a new context where a discrete, entire task can be performed.
Toast: similar to an alert, but displays unintrusive, non-critical messages or notifications. Often displayed in a corner of app UI.
Same differences as “alert”, though toasts may not be live regions. Toasts are very likely to disappear on timeout.
Tooltip: A simple text label supporting another bit of UI.
Popups support fairly arbitrary contents. Tooltips, on the other hand, should only have text contents because they 1) can be used in calculations of the accessible name or description of the element they are attached to and 2) moving hover and/or focus away from an element typically dismisses its tooltip. If there are any interactive elements in the tooltip, they will be unreachable by some users, including keyboard-only users or some people with motor disabilities.