Rhythm - together with pitch (frequency), dynamics (volume), and timbre (tone) - is one of the fundamental components of music.
In an abstract sense, rhythm is the pattern of sounds and silence over time, and more generally is the aspect of music most prominently associated with the time dimension. But rhythm is neither exclusively concerned with time nor the only time-related component of music. Rhythm tells you when each note is played, for how long, and with what emphasis.
Tempo is the speed at which a composition is played; how fast or slow a piece is.
In contemporary music, tempo is often specified as beats-per-minute (BPM), the steady "pulse rate" that underpins the basic rhythmic structure of a song. While not quite universally true, it's not too far off to think of each "beat" as a quarter note, and hence of BMP as the number of quarter notes (or quarter-note equivalents) to be played 60 seconds.
Traditionally, however, a musical score would describe the tempo of a composition is slightly vaguer terms such as allegro for "fast" or lento for "slow". But each of these tempo markings corresponds to a small range of BPM values, by convention. (See this post for examples of the bpm equivalent to traditional tempo annotations.)
Meter is the rhythmic structure of a composition, how the beats are organized into phrases, bars, measures and groups.
The time signature is the most prominent representation of meter in the conventional Western musical notation. These annotations - the stacked numbers that look like a fraction without the dividing line that appear next to the clef symbol at the beginning of the score - specify (a) the number of beats per measure (via the top number) and (b) the note type (duration) which represents a single beat (via the bottom number). For example, the common 4 / 4 time indicates four beats per measure, where each beat is represented by a quarter note. The waltz time signature of 3 / 4 indicates three quarter notes per measure. The compound time signature 6 / 8 specifies six beats per measure, where each beat is represented by an eighth note.
But meter is more than the number of beats per measure. Meter also reflects how beats are grouped and accented within a measure. The traditional 4 / 4 rock rhythm, for example, places an emphasis on the "downbeats" (first and third) by counting "ONE two three four."
This grouping is often implied by the time signature. Measures in 6 / 8 time, for example, are conventionally subdivided into triplets - two groups of three beats that are counted as "ONE-two-three four-five-six" rather than "ONE-two three-four five-six" or simply "ONE two three four five six".
But the full meter is not always explicit or unambiguous based on the time signature alone. A "backbeat" meter, for example, is also specified as 4 / 4 but emphasizes the "upbeats" (beats 2 and 4) instead.
Meter can also reflect patterns that span more than one measure. The 12-bar blues structure may represent a rhythmic as well as a melodic (chord progression) pattern. The way in which a singer (or guitarist) breaks up melody in multi-bar "phrases" is another aspect of song's meter.
Articulation reflects the dynamics within each beat (or note) and the transition between them.
For example a staccato rhythm is characterized by short, truncated notes with a bit of a gap between each beat. A legato rhythm is the opposite - beats are connected, even slurred, and smoothly transition from one note to the next.
Changing the mechanics of the performance - like when a violinist switches from bowing to plucking, or a guitarist executes a palm-mute, or lets a particular note ring for emphasis - can be another example of articulation.
A rhythm guitarist's upstroke/downstroke strumming pattern could also be considered a type of articulation.
Other Aspects of Rhythm
Tempo, meter and articulation are powerful concepts - and a conventional way to model rhythm in formal music theory - but not as pure or complete of a taxonomy as we might hope.
These aren't quite independent, orthogonal dimensions. There's some overlap between the concepts: notably there is some obvious interplay between meter and articulation when it comes to syncopation, accents and emphasis. There's even overlap with some of the other fundamental components of music: the line between rhythmic articulation and melodic dynamics and timbre is fuzzy.
These dimension aren't truly comprehensive, either. There are aspects - widely used aspects - of rhythm that aren't well represented by these concepts alone. Swing rhythms that assign slightly different durations to each beat are one example. Triplet feel (a related concept) is another.
Like a lot of topics in music theory and musical notation, mathematical purity or precision isn't really the intent. A written score isn't a recording on paper. It's more of a collaboration between composer and performer. Music is an interpretive process. You're meant to bring some of yourself to the performance.