How can I see how busy the cluster is?

You can get the current status of all nodes on a cluster using the command sinfo, you will get a printout like the following.

The nodelist column has been truncated for readability

PARTITION AVAIL JOB_SIZE TIMELIMIT    CPUS S:C:T    NODES   STATE    NODELIST
large*    up    1-infini 3-00:00:00     72 2:18:2       1   down*      wbn128
large*    up    1-infini 3-00:00:00     72 2:18:2     133   mixed      wbn[009-020...
large*    up    1-infini 3-00:00:00     72 2:18:2       7   allocated  wbn[031,038
large*    up    1-infini 3-00:00:00     72 2:18:2      85   idle       wbn[021,037...
long      up    1-infini 21-00:00:0     72 2:18:2      64   mixed      wbn[009-020...
long      up    1-infini 21-00:00:0     72 2:18:2       5   allocated  wbn[031,077...
gpu       up    1-infini 3-00:00:00      8  8:1:1       1   reserved   vgpuwbg004
gpu       up    1-infini 3-00:00:00      8  8:1:1       3   idle       vgpuwbg[001-003]
igpu      up    1-infini 3-00:00:00      8  8:1:1       1   reserved   vgpuwbg004
prepost   up    1-infini    3:00:00     72 2:18:2       2   down*      wbl[003,005]
prepost   up    1-infini    3:00:00     72 2:18:2       2   mixed      wbl[002,010]
prepost   up    1-infini    3:00:00     72 2:18:2       5   allocated  wbl[001,004...
bigmem    up    1-infini 7-00:00:00     72 2:18:2       1   down*      wbl003
bigmem    up    1-infini 7-00:00:00     72 2:18:2       2   mixed      wbl[002,010]
bigmem    up    1-infini 7-00:00:00     72 2:18:2       5   allocated  wbl[001,004...
hugemem   up    1-infini 7-00:00:00    128 4:16:2       1   mixed      wbh001

Each partition has a row for every state it's nodes are currently in.

For example, the large partition currently has  1 down node,  133 mixed nodes,  7 allocated nodes and  85 idle nodes.

The most common node states you are likely to see are:

State	Description
idle	All CPUs on this node are unallocated and available for use.
allocated	All CPUs on this node are currently allocated.
mixed	Some CPUs on this node are unallocated, smaller jobs are likely to land here.
down	The node is unavailable for use
reserved	This node has been reserved, and is only available for some users (in the case of the igpu partition, please contact NeSI support if you wish to use it).
draining	Jobs are currently running on this node, but is not available for new jobs.

A full list of node states can be found here.

If you are interested in the state of one partition in particular you may want to use the command squeue -p <partition> to get the current queue of the partition <partition>.