der Rutsch
slide
A short, gliding movement on a smooth or sloping surface.
Nach dem Regen war der Rutsch die nasse Wiese hinunter besonders schnell.
After the rain, the slide down the wet meadow was especially fast.

slide / slip
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slide
A short, gliding movement on a smooth or sloping surface.
Nach dem Regen war der Rutsch die nasse Wiese hinunter besonders schnell.
After the rain, the slide down the wet meadow was especially fast.

The noun 'der Rutsch' (slide/slip) is derived directly from the verb 'rutschen' (to slide). It describes the action or the result of sliding.
Die Kinder hatten Spaß bei jedem Rutsch auf der Eisfläche.
The children had fun with every slide on the ice.
The expression 'einen guten Rutsch' is used exclusively on New Year's Eve to wish someone a good transition into the new year. People often respond with 'Danke, ebenfalls!' (Thanks, same to you!).
Ich wünsche dir und deiner Familie einen guten Rutsch ins neue Jahr!
I wish you and your family a Happy New Year!
In the literal meaning, 'der Rutsch' (slide) is often used with prepositions that indicate a direction or a place. Typical phrases are 'ein Rutsch auf' (a slide on), 'ein Rutsch in' (a slide into), or 'ein Rutsch über' (a slide across/over).
Mit einem lauten Platscher endete sein Rutsch im Wasser.
With a loud splash, his slide ended in the water.
When 'Rutsch' is used in connection with mountains, slopes, or blocked roads, it usually means a landslide. In this context, it is a shorter form of 'Erdrutsch' (landslide) or 'Bergrutsch' (mountain landslide).
Wegen eines Rutschs war die Passstraße für mehrere Stunden gesperrt.
Because of a landslide, the pass road was closed for several hours.
slip / lapse
'der Rutsch' and 'der Ausrutscher' are both connected with sliding on a slippery surface, but they are used differently. 'Ein Rutsch' is the short sliding movement itself. The word also appears in the fixed expression 'einen guten Rutsch', used on New Year's Eve to wish someone a good start to the new year. 'Ein Ausrutscher' is usually an unintended slip, often almost leading to a fall. Figuratively, 'Ausrutscher' can also mean a small mistake or a thoughtless statement. The words can be confused because of their similar form. The important point is: der Rutsch is the movement or the New Year's Eve greeting; der Ausrutscher is a mishap or a mistake.

Am Ende der Feier wünschten wir allen einen guten Rutsch.
At the end of the party, we wished everyone a Happy New Year.
Auf dem nassen Boden hatte Tom einen Ausrutscher, fiel aber nicht hin.
Tom slipped on the wet floor, but he did not fall down.