You may have noticed that I like to make cards that are a bit different to the norm and often need to glue pieces of card together to make dimensional shapes. My favourite glue for this sort of construction is PVA. Once set, it is not going to come apart. I just use a cheap 99p for 118ml pot from Dunelm Mill branded "Craft Factory" which has a great spreader and stand so you can store it with the nozzle at the bottom of the pot.
Many of the more expensive branded glues are just PVA diluted with water to a greater or lesser extent. I also use PVA for matting and layering, taking care not to spread too much glue (you just want a thin smear).
Ranger Inkssentials Glossy Accents
Occasionally, for example when constructing the "pages" in a book card, I want a glue that sets more quickly. Then I will reach for my bottle of Glossy Accents. Glossy accents would suffice for most of the times when PVA is used above, but it is a lot more expensive at approx £5 for 68ml.
Pinflair Dimensional Glue Gel
When a bit of dimension is required, for example if you want to raise an image from a background, little blobs of Pinflair dimensional glue are ideal. I also use Pinflair for attaching bulky embellishments such as charms to cards, and for constructing some paper flowers. Pinflair can take a good while to set, especially if you are using large quantities, so set aside projects to dry overnight. I use Pinflair rather than a cheaper silicone glue mostly because it doesn't smell bad. You can also get a syringe to use with the glue making finer applications possible.
Claudine Hellmuth Matt Multi-Medium
This is fabulous when you know you are going to end up with glue everywhere and you don't want it to show! I use it a lot for sticking detailed diecuts to collages, and also for tissue paper. Any excess glue dries matt so it doesn't notice. You can also use matt multi medium for transfer techniques (transferring laser-printed images onto canvas for example). The glue dries very quickly and is best applied with an old paintbrush (I keep one just for use with glue).
You need a speciality glue for certain techniques such as for use with gilding flakes (something like IndigoBlu's "FlitterGlu" for example). This type of glue stays sticky for a long time. I also use a speciality glue for stamping and adhering glitter. There are various types, such as embossing powder glues, but my favourite is a glue pad.
Foam tape is useful for raising large areas such as images when matting and layering, and is also invaluable for frame cards and shaker card frames. Be aware though that the adhesive on tapes can dry out, so it may not be a permanent bond.
Double-sided (ds) tape
Many people use ds tape and swear by it, but in my experience it dries out and can cause your cards to fall apart. It may be the case that I have just been unlucky with the brands I've used, but I never risk it any more.