I use the following:

R version 3.5.1 (2018-07-02)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 18.04.1 LTS
Bioconductor 3.8, BiocManager 1.30.4

 I have tried to install scater in a multiple of different ways now, and I always end up with a variant of the following warning message:

Installing package(s) 'scater'
The downloaded source packages are in
'/tmp/RtmpE6nZpa/downloaded_packages'
There were 13 warnings (use warnings() to see them)
>> warnings()
Warning messages:
12: In install.packages(pkgs = doing, lib = lib, repos = repos, ... :
installation of package 'scater' had non-zero exit status

I have tried to install through https and http,and got either the warning message from above or a problem with connecting to the SSL (when using https, sorry I forgot to save that error message).

source("https://bioconductor.org/biocLite.R
biocLite("scater")

 I also tried to install it as recommended from your website: https://www.bioconductor.org/install/#install-bioconductor-packages but recieved the same warnings.

>>if (!requireNamespace("BiocManager"))
>>install.packages("BiocManager")
>> BiocManager::install()
The downloaded source packages are in
'/tmp/RtmpE6nZpa/downloaded_packages'
Warning message:
In install.packages(update[instlib == l, "Package"], l, contriburl = contriburl, :
installation of package 'edgeR' had non-zero exit status
>> install.packages("edgeR")
Installing package into '/usr/local/lib/R/site-library'
(as 'lib' is unspecified)
Warning message:
package 'edgeR' is not available (for R version 3.5.1)
>> BiocManager::install(c("scater"))
Bioconductor version 3.8 (BiocManager 1.30.4), R 3.5.1 (2018-07-02)
Installing package(s) 'scater'
The downloaded source packages are in
'/tmp/RtmpE6nZpa/downloaded_packages'
There were 13 warnings (use warnings() to see them)

And finally, i checked if scater existed at all, and found the following:

>>BiocManager::available()
[13831] "SCAT"
[13832] "scatr"
[13833] "scatterD3"
[13834] "scatterpie"
[13835] "scatterplot3d"

What is this supposed to mean? Have scater changed name, or is it not available any more?

scater  is still available and installable from Bioconductor.

Your choice of what to show from the installation log is unfortunate as you've removed all the relevant error messages. My guess is that your installation directory (/usr/local/lib/R/site-library) is not writeable by you (as an ordinary user). Normally, in such cases, R asks if you want to install into a personal library tree that you should have write access to, but this might also not be working for various reasons.

The other typical reason for widespread package installation failures is if your compiler lacks C++11 support, but I find this hard to believe for such a recent version of Ubuntu - assuming you're not using some custom build environment involving conda or brew that might mess up ABI compatibilities.